LRFD Bridge Design Fundamentals and Applications By Tim Huff CRC Press 2022

LRFD Bridge Design
LRFD Bridge Design
Fundamentals and Applications
Tim Huff
First edition published 2022
by CRC Press
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and by CRC Press
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© 2022 Tim Huff
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Library of Congress Cataloging‑in‑Publication Data
Names: Huff, Tim, author.
Title: LRFD bridge design : fundamentals and applications / Tim Huff,
Tennessee Technological University Cookeville, Tennessee.
Description: First edition. | Boca Raton : CRC Press, [2022] | Includes
bibliographical references and index.
Identifiers: LCCN 2021043502 (print) | LCCN 2021043503 (ebook) | ISBN
9781032208367 (hardback) | ISBN 9781032208374 (paperback) | ISBN
9781003265467 (ebook)
Subjects: LCSH: Bridges--Design and construction--Textbooks. | Load factor
design--Textbooks.
Classification: LCC TG300 .H84 2022 (print) | LCC TG300 (ebook) | DDC
624.2/5--dc23
LC record available at https://lccn.loc.gov/2021043502
LC ebook record available at https://lccn.loc.gov/2021043503
ISBN: 978-1-032-20836-7 (hbk)
ISBN: 978-1-032-20837-4 (pbk)
ISBN: 978-1-003-26546-7 (ebk)
DOI: 10.1201/9781003265467
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by Deanta Global Publishing Services, Chennai, India
Access the Support Material at www.routledge.com/9781032208367
Contents
Preface.......................................................................................................................ix
Acknowledgements....................................................................................................xi
Author biography.................................................................................................... xiii
Chapter 1
Introduction...........................................................................................1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Chapter 2
Loads on Bridges................................................................................. 23
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
Chapter 3
Dead Loads (DC and DW).......................................................24
Live and Impact Loads (LL and IM).......................................25
Braking Forces (BR)................................................................. 27
Centrifugal Forces (CE)........................................................... 27
Wind Loads (WS and WL)....................................................... 29
Collision Loads (CT and CV).................................................. 30
Temperature Loads (TU).......................................................... 32
Earthquake Loads (EQ)........................................................... 33
Water Loading (WA)................................................................ 43
Solved Problems.......................................................................46
Exercises...................................................................................60
Load Combinations and Limit States.................................................. 63
3.1
3.2
Chapter 4
The Project Bridge......................................................................1
Preliminary Dimensions............................................................. 5
Bridge Girder Behavior at Various Stages of Construction.........7
Bridge Materials.........................................................................9
Software for Bridge Engineering............................................. 13
Section Properties..................................................................... 14
Solved Problems....................................................................... 14
Exercises...................................................................................20
Solved Problems....................................................................... 68
Exercises...................................................................................80
Deck and Parapet Design.................................................................... 83
4.1
4.2
4.3
4.4
4.5
Parapet Design.......................................................................... 83
Deck Overhang Design............................................................. 85
Interior Bay Deck Design......................................................... 85
Solved Problems....................................................................... 88
Exercises................................................................................. 106
v
vi
Chapter 5
Contents
Distribution of Live Load.................................................................. 107
5.1
5.2
5.3
5.4
5.5
Chapter 6
AASHTO Equations............................................................... 108
The Lever Rule....................................................................... 111
Rigid Cross-Section Method.................................................. 112
Solved Problems..................................................................... 113
Exercises................................................................................. 124
Steel Welded Plate I-Girders............................................................. 125
6.1
Flexural Resistance at the Strength Limit State..................... 126
6.1.1 Composite Compact Sections in Positive Flexure........ 126
6.1.2 Non-Compact Composite Sections in
Positive Flexure......................................................... 128
6.1.3 Negative Flexure and Non-composite Sections........ 129
6.2 Shear Resistance..................................................................... 131
6.3 Transverse Stiffener Design................................................... 132
6.4 Bearing Stiffener Design........................................................ 133
6.5 Fatigue Design........................................................................ 134
6.6 Field Splice Design................................................................. 137
6.7 Stability Bracing..................................................................... 142
6.8 Shear Studs............................................................................. 146
6.9 Plastic Moment Computations................................................ 148
6.10 Solved Problems..................................................................... 148
6.11 Exercises................................................................................. 174
Chapter 7
Precast Prestressed Concrete Girders............................................... 179
7.1
7.2
7.3
7.4
7.5
7.6
Stress Analysis........................................................................ 179
Flexural Resistance................................................................ 180
Shear Resistance..................................................................... 182
Continuity Details.................................................................. 184
Mild Tensile Reinforcement in Girders.................................. 186
Negative Moment Reinforcement for Girders
Made Continuous.................................................................... 187
7.7 Transfer and Development Length......................................... 190
7.8 Stress Control Measures......................................................... 190
7.9 Solved Problems..................................................................... 191
7.10 Exercises................................................................................. 210
Chapter 8
Bridge Girder Bearings..................................................................... 215
8.1
8.2
8.3
8.4
8.5
8.6
Elastomeric Bearings.............................................................. 215
Steel Assembly Bearings........................................................ 220
Isolation Bearings................................................................... 222
Anchor Rods........................................................................... 227
Solved Problems..................................................................... 227
Exercises................................................................................. 239
vii
Contents
Chapter 9
Reinforced Concrete Substructures................................................... 241
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
9.10
9.11
9.12
9.13
9.14
Pier Cap Design...................................................................... 242
Pier Column Design............................................................... 245
Spread Footing Design...........................................................246
Pile Cap Design...................................................................... 247
Drilled Shaft Design............................................................... 250
Pile Bent Design..................................................................... 251
Bridge Pier Displacement Capacity under
Seismic Loading..................................................................... 255
The Alaska Pile Bent Design Strategy................................... 258
Concrete Filled Steel Tubes (CFST)....................................... 258
9.9.1 CFST Design in Accordance with BDS
Sections 6.9.6 and 6.12.2.3.3..................................... 259
9.9.2 CFST Design by BDS Sections 6.9.5 and
6.12.3.2.2 and GS Section 7.6....................................260
9.9.3 Steel Tube Design without Concrete Fill.................. 262
9.9.4 CFST Design for Extreme Event Limit States.......... 263
Two-Way Shear....................................................................... 265
Fatigue Related Issues in Reinforced Concrete...................... 265
Abutment Design....................................................................266
Solved Problems..................................................................... 269
Exercises.................................................................................302
Chapter 10 Seismic Design of Bridges................................................................ 305
10.1
10.2
10.3
10.4
10.5
10.6
10.7
10.8
Force-based Seismic Design by the LRFD BDS...................306
Displacement-based Seismic Design by the LRFD GS.........308
Capacity Design Principles.................................................... 311
Ground Motion Selection and Modification for Response
History Analysis..................................................................... 313
Substitute-Structure Method (SSM) Analysis........................ 317
Shear Resistance at the Extreme Event Limit State............... 320
Solved Problems..................................................................... 322
Exercises.................................................................................340
Chapter 11 Seismic Isolation of Bridges.............................................................. 343
11.1 Partial Isolation of Interstate 40 over State Route 5............... 343
11.2 Seismic Retrofit of Interstate 40 over the
Mississippi River.................................................................... 353
11.3 Solved Examples.................................................................... 353
11.4 Exercises.................................................................................360
Bibliography.......................................................................................................... 363
Index....................................................................................................................... 367
Preface
Beginning with basic concepts in bridge geometry, the text progresses through discussions on the various elements of typical I-girder bridges. Criteria from the 9th
edition of the American Association of State Highway and Transportation Officials
(AASHTO) LRFD Bridge Design Specifications are presented and applied to sample
problems. Many examples are based on constructed bridges designed by the author.
Steel and concrete I-girder design, deck and parapet design, load calculations, bearing design, and substructure design are all included. The book ends with chapters
devoted specifically to seismic design and seismic isolation applied to bridges. Each
chapter ends with a section of solved problems to illustrate the principles covered,
followed by exercises which may be used as exam problems by instructors. An
Appendix consists of detailed solutions for the exercises.
This book is intended for use as a reference for practicing bridge engineers and as
a textbook for a course (or multiple courses) in bridge engineering.
Tim Huff is a faculty member of the Civil & Environmental Engineering
Department at Tennessee Technological University in Cookeville, where he resides
with his beautiful and talented wife, Monica, an artist and a teacher.
This book is dedicated to my family – Monica, Majo, Esteban, Troy, Holli, and
my parents, Bill and Sue. My inspiration for the book is my students at Tennessee
State University and Tennessee Tech University, as well as the experiences encountered over the course of 35 years as a practicing structural engineer.
Blessings and peace to all, without qualification. May you be happy and healthy,
free from pain and suffering, and may you find joy and peace always.
ix
Acknowledgements
I thank my mentors for their confidence in me, my co-workers for their friendship,
my students for teaching me more than I taught them. I thank my family for love and
inspiration.
The Bridge Builder
An old man going a lone highway,
Came, at the evening cold and gray,
To a chasm vast and deep and wide,
Through which was flowing a sullen tide.
The old man crossed in the twilight dim;
The sullen stream had no fear for him;
But he turned when safe on the other side
And built a bridge to span the tide.
“Old man,” said a fellow pilgrim near,
“You are wasting your strength with building here;
Your journey will end with the ending day;
You never again must pass this way;
You have crossed the chasm, deep and wide,
Why build this bridge at eventide?”
The builder lifted his old gray head:
“Good friend, in the path I have come,” he said,
“There followeth after me today
A youth whose feet must pass this way.
This chasm that has been as naught to me
To that fair-haired youth may a pit-fall be.
He, too, must cross in the twilight dim;
Good friend, I am building the bridge for him.”
Will Allen Dromgoole
xi
Author biography
Tim Huff has 35 years of experience as a practicing structural engineer. Dr Huff
has worked on building and bridge projects in the United States and has contributed
to projects in India, Ethiopia, Brazil, the Philippines, and Haiti as a volunteer structural engineer with Engineering Ministries International. He is a faculty member
of the Civil & Environmental Engineering Department at Tennessee Technological
University in Cookeville, where he resides with his beautiful and talented wife,
Monica, an artist and teacher.
xiii
1
Introduction
This course is intended for senior level undergraduate civil engineering majors with
an emphasis on structural engineering, and graduate level structural engineering students. Practitioners may also find the material to be a valuable reference. After this
introductory chapter (Chapter 1) on preliminary design considerations and construction stages in the life of a bridge, subsequent topics covered include the following:
•
•
•
•
•
•
•
•
•
load calculations and limit states: Chapters 2 and 3
bridge deck design: Chapter 4
live load distribution: Chapter 5
steel welded plate girders: Chapter 6
prestressed concrete girder bridges: Chapter 7
bridge girder bearings: Chapter 8
reinforced concrete substructures and foundation design: Chapter 9
seismic design concepts for bridges: Chapter 10
seismic isolation applied to bridges: Chapter 11
Load and resistance factor design (LRFD) of I-girder-type bridges is the basis of the
superstructure-related content in Chapters 5 through 8. Primary references for bridge
design include specifications from the American Association of State Highway and
Transportation Officials (AASHTO). These documents, along with the shorthand
notation used for each in this book, are summarized below.
• AASHTO LRFD Bridge Design Specifications, 9th edition (AASHTO,
2020). Hereafter referred to as the LRFD BDS.
• AASHTO Guide Specification for LRFD Seismic Bridge Design, 2nd edition
(AASHTO, 2011). Hereafter referred to as the LRFD GS.
• AASHTO Guide Specification for Seismic Isolation Design, 4th edition
(AASHTO, 2014). Hereafter referred to as the GS ISO.
• AASHTO LRFD Bridge Construction Specifications, 4th edition (AASHTO,
2017). Hereafter referred to as the LRFD BCS.
1.1 THE PROJECT BRIDGE
The Project Bridge, defined by Figures 1.1​​ ​​through 1.7, provides the basis for much
of the discussion on the design of the various elements of a typical I-girder bridge.
Some of the examples in each chapter are based on this Project Bridge. Consisting
of two 90-ft spans, the 34-ft 5-inch- wide Project Bridge design will include both
prestressed concrete I-girder and steel I-girder superstructure options. Deck design,
DOI: 10.1201/9781003265467-1
1
2
LRFD Bridge Design
FIGURE 1.1
Bridge cross section.
FIGURE 1.2
Bridge framing plan (steel girder option).
FIGURE 1.3
Plan view – intermediate diaphragms (concrete girder option).
stability bracing (for the steel option), bearing design, bent (pier) design, and foundation design will all be explored.
Figures 1.3 through ​​1.6 for the concrete girder option were generated using the
LEAP Bridge Concrete software by Bentley. With prestressed concrete I-girder
bridges, intermediate diaphragms, shown at the one-third span points in Figure 1.3,
Introduction
FIGURE 1.4
3
Isometric of pier.
are often provided to improve stability during construction. It is not uncommon
for these diaphragms to have “standard” details with little or no design provisions.
Precast, prestressed I-girders typically span from support to support and behave as
simply supported beams until the cast-in-place deck attains design strength.
Figure 1.2 for the steel I-girder option depicts cross-frames between the girders spaced 30 ft apart in the positive moment (compression in the top of the deck)
region and at 15 ft in the negative moment region. These cross-frames become
critical elements for stability, particularly during construction, for long span steel
I-girder bridges. Careful attention to design requirements becomes necessary in such
cases. Figure 1.7 for the steel I-girder alternate identifies field splice locations. Steel
I-girders constructed in such a manner behave as continuous beams for all loads,
though part of the load is carried by the non-composite section and part by the composite section. More discussion on continuity will be reserved for later. Each field
section is limited by shipping and handling requirements and varies from place to
place, and among fabricators and contractors.
A brief discussion on the general behavior of each major element is in order, prior
to developing detailed design requirements.
First, consider the parapet in Figure 1.1. This element behaves much as a cantilever beam subjected to primarily lateral loads. Certainly, wind loads act on the
4
LRFD Bridge Design
FIGURE 1.5
Basic pier dimensions (preliminary).
FIGURE 1.6
Isometric of entire bridge.
Introduction
FIGURE 1.7
5
Elevation view of bridge (steel girder option).
parapet. But the primary loading affecting parapet details is a vehicular collision
load, typically assumed to be a lateral load applied at the top of the cantilever parapet. The load for which a parapet must be designed will hopefully never occur.
Second, consider the typically cast-in-place concrete deck, which behaves as a continuous beam between I-girders. The deck will experience tension in the top fiber
(negative moment) over the supports (the I-girders) and tension in the bottom fiber
(positive moment) midway between the supports. The negative moment condition
requires consideration of two cases: (1) the interior negative moment over a support
and (2) the overhang negative moment produced by the parapet weight, the deck selfweight, any overlay, and any traffic outside the exterior girder. The I-girders themselves will also behave as continuous beams (at least for part of the load) spanning
between adjacent abutments and piers. Whereas these I-girders could be designed
using a complex two-dimensional grillage model, more often the design is completed
with a line-girder analysis of a single girder. This concept will be explored in later
sections, primarily in Chapter 5 on live load distribution. Finally, consider the bent
(pier) in Figure 1.4. This is typically a reinforced concrete rigid frame in the transverse direction. Behavior in the longitudinal direction may be either as a rigid frame
or as a cantilever.
1.2 PRELIMINARY DIMENSIONS
Before design of the various bridge elements may commence, it is necessary to establish the cross-section geometry in terms of girder spacing, ‘S’, and overhang (cantilever) dimension, ‘C’ (see Figure 1.1). Although it is theoretically acceptable to use
larger overhangs, it is usually most economical, in terms of balancing deck moments
and interior/exterior girder live load distribution, to use a value for ‘C’ of no more
than 40% of ‘S’.
Though not a requirement, it seems most advantageous to provide a girder spacing of 8 ft to 10 ft for prestressed girders, and 9 ft to 12 ft for steel I-girders, based
6
LRFD Bridge Design
TABLE 1.1
Estimated Superstructure Depth
Approximate Depth of Superstructure
Superstructure
Simple Spans
Continuous Spans
0.045L
0.040L
Steel I-Girder + Deck
0.040L
0.032L
I-Girder Portion of Steel I-Girder
0.033L
0.027L
Prestressed Concrete
I-Beams + Deck
on experience. Certainly, girder spacing outside these ranges has been adopted successfully on many projects.
Required superstructure depths, Dss, may be estimated using Table 1.1, based on
Section 2.5 in the AASHTO LRFD BDS. In the tabulated expressions, ‘L’ is the
span length. The table is a rough guide, not a strict requirement, and may be used to
establish preliminary superstructure depths. Often, the final design depth will not
satisfy the tabular estimates.
The estimates are traditionally recommended (as opposed to required) minimum
values. Nonetheless, they may prove valuable in establishing a starting point for
required girder depth. Later, for both the concrete and steel girder options, final
selected dimensions will be larger for the Project Bridge than these computed, recommended, minimum values. The primary consideration in superstructure depth
is vertical clearance. Bridges over highways typically require a minimum vertical
clearance of 16 ft from the roadway below to the bottom of the girder. To allow for
future overlay of the road below, it is common practice to provide 16 ft 6 inches of
vertical clearance (see the LRFD BDS, Section 2.3.3.2). For river and stream crossings, vertical clearance is a design consideration for the hydraulic properties of the
structure and the low girder elevation. The vertical clearance requirement for bridges
is 23 ft, as governed by the Manual for Railway Engineering (AREMA, 2021).
Horizontal clearance is important in the placement of substructures. A 30-ft clear
zone is typically adopted for high-volume roadways. The clear zone is the “unobstructed, traversable area provided beyond the edge of the through-traveled way for
the recovery of errant vehicles” (AASHTO, 2011). The clear zone is generally measured from the edge of the outermost lanes, though there are exceptions. Horizontal
clearance is measured perpendicular to the roadway below. Skewed bridges thus
require longer spans to satisfy horizontal clearance requirements. Navigable waterways often require very long spans to accommodate horizontal clearance requirements for water traffic in the form of barges and ships.
Terminology for intermediate supports varies. End supports for bridges are called
abutments. Intermediate supports are called either piers or bents. One frequently
used distinction is to use the term “pier” for water crossings and “bent” for roadway
and railway crossings. The two terms, pier and bent, will be used interchangeably
in this book.
Introduction
7
1.3 BRIDGE GIRDER BEHAVIOR AT VARIOUS
STAGES OF CONSTRUCTION
Although there are exceptions, prestressed girder bridges are typically constructed
so as to behave as follows:
• non-continuous and non-composite for self-weight, deck weight, intermediate diaphragms, and construction live loads
• continuous and composite for parapets, sidewalks, future overlay, and traffic
Once again, there are exceptions, but steel I-girder bridges, on the other hand, are
typically designed and constructed so as to behave:
• continuous and non-composite for self-weight, deck weight, cross frames,
lateral bracing, and construction live loads
• continuous and long-term composite for parapet, sidewalks, future overlay,
and utilities
• continuous and short-term composite for traffic
Long-term composite properties for steel I-girders are computed using steel as
the primary material with a modular ratio of 3n applied to concrete. Short-term
composite properties for steel I-girders are computed using a modular ratio,
n = E S/E C (ratio of steel to concrete Young’s modulus). The long-term property
calculations are intended to account for creep under permanent loads, while
research has shown that short-term properties are more appropriate for transient
loads.
For prestressed girders, continuity is achieved by extending strands beyond the
beam end at the pier, bending the strands up, and embedding the strands into a castin-place diaphragm. These bent-up strands are not shown in Figure 1.8 and will
be discussed in later sections. Prestressed concrete girder continuity also requires
FIGURE 1.8
Schematic of a typical prestressed concrete girder bridge.
8
LRFD Bridge Design
relatively heavy longitudinal reinforcement in the deck to resist negative moments at
interior supports.
Figures 1.8 and 1.9 are schematic depictions of typical prestressed concrete and
steel I-girder bridges, respectively. Figures 1.10 and 1.11 are photographs of actual
structures under construction.
FIGURE 1.9
FIGURE 1.10
Schematic of a typical steel I-beam bridge.
Demonbreun Street Viaduct in Nashville, TN, under construction.
9
Introduction
FIGURE 1.11
SR-70 over Center Hill Lake in Smithville, TN, under construction.
1.4 BRIDGE MATERIALS
Properties for some of the reinforcing steels commonly used in bridge construction
are summarized in Table 1.2. A 706 reinforcement has a higher reliable ultimate
strain than does A 615 reinforcement, and has a maximum cap on yield strength,
both characteristics being beneficial in the seismic design of ductile elements.
A 706 reinforcement is required in plastic hinging regions for longitudinal reinforcement of columns in regions of high seismic hazard (see Section 8.4.1 of the
LRFD GS). For computing overstrength plastic moment resistance in seismic analysis, the specified magnifier (λmo) is equal to 1.2 for A 706 bars and 1.4 for A 615 bars
(see Section 8.5 of the LRFD GS).
Table 1.3 provides design information for prestressing steels used in bridge construction. Low-relaxation (“low-lax”) strands generally experience less loss of prestress force compared to their stress-relieved counterparts and are a popular selection
for modern bridges. Properties of two of the most frequently encountered 270K,
7-wire strands, whether low-lax or stress-relieved, are:
• ½-inch diameter strands: A = 0.153 in2/strand
• 0.6-inch diameter strands: A = 0.217 in2/strand
Standard AASHTO I-beam and Bulb-T girders are depicted in Figures 1.12 and
1.13, respectively, with properties listed in Tables 1.4 through ​1.6. The figures are
10
LRFD Bridge Design
TABLE 1.2
Reinforcing Steels
εcl
εtl
εSU
εSUR
90
0.0020
0.0050
0.09 #4-#10
0.06 #11-#18
0.06 #4-#10
0.04 #11-#18
100
105
80
0.0028
0.0030
0.0020
0.0050
0.0056
0.0050
0.12 #4-#10
0.09 #11-#18
0.09 #4-#10
0.06 #11-#18
100
0.0030
0.0056
150
0.0040
0.0080
ASTM
Grade
fy, ksi
fu, ksi
fye, ksi
A 615
40
40
60
68
60
60
75
80
60
75
80
60 min
78 max
80 min
98 max
A 706
80
A 1035
100
100
68
TABLE 1.3
Prestressing Steels
Diameter
fpu, ksi
fpy, ksi
E, ksi
Low-relaxation
7-wire strand
0.375–0.600
270
0.90fpu
28,500
Plain bars
0.750–1.375
150
0.85fpu
30,000
Deformed bars
0.625–2.500
150
0.80fpu
30,000
Material
FIGURE 1.12
AASHTO I-beams (PCI, 2014).
11
Introduction
FIGURE 1.13
AASHTO bulb-T beams (PCI, 2014).
TABLE 1.4
Concrete I-beam Dimensions (inches)
Type
D1
D2
D3
D4
D5
D6
B1
B2
B3
B4
B5
B6
I
28
4
0
3
5
5
12
16
6
3
0
5
II
III
IV
V
36
45
54
63
6
7
8
5
0
0
0
3
3
4.5
6
4
6
7.5
9
10
6
7
8
8
12
16
20
42
18
22
26
28
6
7
8
8
3
4.5
6
4
0
0
0
13
6
7.5
9
10
VI
72
5
3
4
10
8
42
28
8
4
13
10
12
LRFD Bridge Design
TABLE 1.5
Concrete I-beam Properties
Area, in2
yb, inchesa
I, in4
w, klf
276
12.59
22,750
0.287
II
III
IV
V
369
560
789
1,013
15.83
20.27
24.73
31.96
50,980
125,390
260,730
521,180
0.384
0.583
0.822
1.055
VI
1,085
36.38
733,320
1.130
Type
I
a
yb is the distance from the bottom of the beam to the centroid of the cross section.
TABLE 1.6
Concrete Bulb-T Beam Properties
Type
H, in.
HW, in.
A, in2
I, in4
yb, in.
w, klf
BT-54
54
36
659
268,077
27.63
0.686
BT-63
63
45
713
392,638
32.12
0.743
BT-72
72
54
767
545,894
36.60
0.799
TABLE 1.7
Bridge Steels
AASHTO
M270
ASTM
Grade
Shapes
Plates
Fy, ksi
A709
36
All
≤ 4”
36
50
All
≤ 4”
50
50W
All
≤ 4”
50
HPS 50W
NA
≤ 4”
50
HPS 70W
NA
≤ 4”
70
HPS 100W
NA
≤ 2.5”
100
HPS 100W
NA
2.5”– 4”
90
Fu, ksi
58
65
70
70
85
110
100
taken from the literature (PCI, 2014). Concrete Young’s modulus in AASHTO is
taken to be equal to Ec = 1,820( f’c)1/2, where both f’c and Ec are in units of kips per
square inch (ksi).
Steel bridge girders are occasionally made from rolled steel sections, but more
often are welded plate girders. Both I-girder and tub-girder sections are used, with
I-girders being the most common type. Bridge steels are summarized in Table 1.7.
Grade 50W is a weathering steel which performs well without paint and is often preferred due to the lower maintenance costs. HPS (high-performance steel) versions of
the steels possess enhanced toughness and weldability.
13
Introduction
TABLE 1.8
Anchor Rod Properties
Grade
36
Fy, ksi
Fu, ksi
Diameter, inches
36
58–80
½–4
55
55
75–95
½–4
105
105
125–150
½–3
Anchor rods are required in bridge structures for bearing attachment to substructures and to resist seismic loads in support diaphragms. Table 1.8 provides properties
for the various grades of the preferred anchor rod material specification, ASTM F1554.
High-strength bolts used in field splices for steel I-girders, in lateral bracing member connections, and in cross-frame member connections include ASTM F3125,
Grade A325 (also known as Group A) and Grade A490 (also known as Group B).
Specified minimum tensile strength, Fu, is 120 ksi for Grade A325 bolts and 150 ksi
for Grade A490 bolts.
Shear studs used on the top flange of steel I-girders to provide composite action
with the concrete deck are required to conform to the requirements of the AASHTO/
AWS D1.5M/D1.5 Bridge Welding Code and are required to have minimum specified yield and tensile strengths, Fy = 50 ksi and Fu = 60 ksi, respectively.
1.5 SOFTWARE FOR BRIDGE ENGINEERING
Software will be used frequently for the material presented in this course. Programs
used will include:
• Response 2000 for concrete section-analysis, Evan Bentz, (www​.ecf​.utoronto​.ca/​~bentz​/thesis​.htm)
• Consec for section analysis, Robert Matthews (structware​.c​om)
• VisualAnalysis for general structural analysis and design, Integrated
Engineering Software (www​.iesweb​.com​/edu/)
• LRFD Simon for steel bridge girder analysis and design, National Steel
Bridge Alliance (www​.aisc​.org​/nsba ​/design​-resources​/simon/)
• LEAP Bridge by Bentley (/www​.bentley​.com ​/en​/products​/product​-line​/
bridge​-analysis​-software​/openbridge​-designer)
The list below summarizes other freely available resources which bridge engineers
may find useful.
• FHWA Steel Bridge Design Handbook (FHWA, 2015)
• NSBA-Splice, an Excel spreadsheet for steel girder field splice design,
National Steel Bridge Alliance
• NSBA Continuous Span Standards, National Steel Bridge Alliance
14
LRFD Bridge Design
• PCI Bridge Design Manual (PCI, 2014)
• WSDOT BridgeLink Bridge Engineering Software, Washington State
Department of Transportation
1.6 SECTION PROPERTIES
The calculation of composite section properties for bridge girders is based on theory
typically learned in courses dealing with mechanics of materials, advanced steel
design, and reinforced concrete, each of which is a pre-requisite to a full appreciation
of this course. Deflection calculations for simple beams is covered in undergraduate
structural mechanics courses. Section property calculations for composite sections
are needed in all cases for modern bridge girders, both steel and prestressed concrete. Exercises for Chapter 1 focus on examples of such calculations, in addition
to girder spacing and overhang issues discussed in the Introduction. For additional
information on the calculation of plastic moments, the reader is referred to Appendix
D6 of the LRFD BDS (AASHTO, 2020).
Recall that section property calculations for cross-sections composed of more
than one material require the selection of a base material. For reinforced concrete
design, concrete is selected as the base material with steel areas multiplied by the
modular ratio (n) to obtain an equivalent concrete cross section. Subsequent stress
calculations for the steel component require amplification by n. With steel plate girders, the reverse approach is adopted. Concrete is transformed into an equivalent steel
area via division by the modular ratio. Subsequent stress calculations for the concrete
components require division by n.
1.7 SOLVED PROBLEMS
Problem 1.1
Establish an approximate girder spacing and approximate girder depth for
both the concrete and steel girder options for the Project Bridge. The deck
thickness is 8.25 inches and the thickness of the haunch (the gap between
the top of the girder and the bottom of the deck) is 1.75 inches.
Problem 1.2
For the concrete option on the Project Bridge, suppose a prestress force
equal to 929 kips is applied to the girder as a compressive axial force at
each end. Select a BT-54 girder and determine the midspan deflection due
to girder self-weight and the applied axial prestress force. The prestress
force is applied 4.53 inches above the bottom of the girder, thus producing
not only an axial force, but also a negative moment (tension in the top of the
girder) due to eccentric application. Assume the girder behaves as a simple
span and is 87-ft 9-in long. Concrete strength is f’c = 7 ksi. Ignore secondorder effects of the axial load on deflections.
Problem 1.3
For the steel girder option of the Project Bridge, determine the composite properties (both short-term and long-term) for the elastic condition for
an interior girder. Determine the plastic moment of the composite section in
positive flexure (deck in compression). Use a W40 × 215 rolled beam girder,
15
Introduction
8.25-inch deck, 1.75-inch haunch (distance from the top of the girder to the
bottom of the deck in this case), and Grade 50 steel. Deck concrete has a
specified minimum 28-day compressive strength, f’c of 5 ksi.
PROBLEM 1.1
TEH
1/2
The following calculation shows that a girder spacing of no less than 9.057 feet is
required to meet the overhang rule-of-thumb for the Project Bridge.
3S + 2C = 34.417 feet
3S + 2 ( 0.4S ) = 34.417¢ ® S = 9.057 feet
Select S = 9 feet 3 inches with a corresponding C = 3 feet 4 inches for the Project
Bridge.
Define the following:
•
•
•
•
(Dss)PPC is the superstructure depth for the prestressed girder option.
(DBM )PPC is the girder depth for the prestressed girder option.
(Dss)SG is the superstructure depth for the steel girder option.
(DBM )SG is the girder depth for the steel girder option.
The superstructure depth is the girder depth plus the haunch depth plus the deck
thickness.
For the precast, prestressed concrete girder option, assume continuous spans
and use the information from Table 1.1.
( Dss )PPC » 0.040 ´ 90 = 3.6 ft = 43 inches
Whereas (Dss)PPC is, strictly speaking, the total depth of (a) the prestressed concrete girder, (b) the haunch, and (c) the deck, experience has shown that the girder
depth alone should be approximately equal to the calculated value for optimal
design of prestressed girders, in the author’s opinion. Without question, the estimate is just that, an estimate, and many successful projects have incorporated
depths both smaller and larger.
PROBLEM 1.1
TEH
2/2
( DBM )PG ³ 43 inches
For the steel I-girder option, two criteria are specified in Table 1.1. Assess both
and choose the larger value as the initial superstructure depth estimate.
For the steel I-girder plus deck criteria:
( Dss )SG » 0.032 ´ 90 = 2.88 ft = 34.6 inches
( DBM )SG = 34.6 - 8.25 - 1.75 = 24.6 inches
16
LRFD Bridge Design
For the steel I-girder alone:
( DBM )SG » 0.027 ´ 90 = 2.43 ft = 29.2 inches ¬ controls
( DBM )SG ³ 29.2 inches ¬ controls
PROBLEM 1.2
TEH
1/2
For a BT-54 girder, obtain the properties from Table 1.6.
A = 659 in 2
I = 268, 077 in 4
yb = 27.63 inches
w = 0.686 klf
The eccentricity of the applied axial force may now be computed.
e = 27.63 - 4.53 = 23.1 inches
The modulus of elasticity for concrete, Ec, is needed for deflection calculations.
EC = 1, 820 7 = 4, 815 ksi
The end moment is equal to the applied force multiplied by the eccentricity. Since
this moment is negative, the deflection resulting from the moment is upward. The
conjugate beam method can be used to determine the deflection due to applied
end moments in a simply supported beam.
D PS =
M PS L2
8EC I
M PS = 929 ´ 23.1 = 21, 460 inches kips
D PS =
PROBLEM 1.2
( 21, 460 ) (87.75 ´ 12 )
8 ( 4, 815 )( 268, 077 )
2
= 2.30 inches­
TEH
2/2
The deflection at midspan due to self-weight, a uniformly distributed load, may be
determined from the conjugate beam method, double integration of the moment
equation, or from standard textbook formulas.
D SW
4
æ 0.686 ö
5ç
87.75 ´ 12 )
(
÷
5wL4
12 ø
=
= è
= 0.71 inches ¯
384 EC I
384 ( 4, 815 )( 268, 077 )
17
Introduction
The net deflection is the sum of that due to eccentricity of prestress and selfweight. The net deflection is found to be upward.
DTOT = 2.30 inches ↑ + 0.71 inches ↓ = 1.59 inches ↑
PROBLEM 1.3
TEH
1/4
18
PROBLEM 1.3
LRFD Bridge Design
TEH
2/4
19
Introduction
PROBLEM 1.3
TEH
3/4
20
LRFD Bridge Design
PROBLEM 1.3
TEH
4/4
1.8 EXERCISES
E1.1.
A three-span continuous, 129-ft 3-inches-wide interstate bridge consists of
95-ft end spans with a 156-ft center span for a total length of 346 feet.
Estimate the girder depth required if (a) prestressed concrete girders are
used and (b) steel I-girders are used. Also, determine the number of beams
and beam spacing for each girder type.
For the prestressed girder option, use a girder spacing between 8 feet and
10 feet. For the steel I-girder option, use a girder spacing between 9 feet and
12 feet. Use a deck thickness of 8.25 inches.
E1.2.
A five-span continuous, 34-ft-wide bridge consists of 270-ft end spans and
three 335-ft interior spans for a total length of 1,545 feet. Establish a preliminary cross-section (number of girders, girder spacing, and overhang
dimension) configuration and steel I-girder depth for the bridge. To minimize the number of girders, use a target girder spacing of 12 feet. Deck
thickness is 9 inches.
Introduction
E1.3.
A 103-ft long BT-72 prestressed concrete girder consists of forty ½inch diameter 270K-Low-Lax strands. The girder concrete strength is
f’ci = 6,000 psi (initial strength, at release of strands). The centroid of the
strand group is located 10.8 inches above the bottom of the beam. The initial pull on the strands is 75% of the tensile strength. Treat the BT-72 as
a simply supported beam with self-weight, applied end compressive loads
(due to the prestress force), and applied end moments (due to the eccentricity of the prestress force). This is representative of the condition for the
completed beam before installation on a bridge. While the strand stress is
zero at the ends and builds up to the initial pull a development length from
the ends, treat the prestress force as being applied at the ends. Additionally,
the strand stress experiences elastic shortening immediately and the effective prestress force would be somewhat less than 0.75fpu. For this academic
problem, ignore the effects of strand development and prestress loss.
Determine the stresses (and identify whether they are tension or compression stresses) at midspan in the top and bottom of the BT-72. If the
stress limits are 0.65f’ci in compression and 0.24(f’ci)1/2 in tension, does the
girder satisfy stress limits at midspan?
Determine the stresses (and identify whether they are tension or compression stresses) at the girder end in the top and bottom of the BT-72. If the
stress limits are 0.65f’ci in compression and 0.24(f’ci)1/2 in tension, does the
girder satisfy stress limits at midspan?
Determine the deflection (and identify whether it is upward or downward) at midspan. Use a first-order analysis (do not consider the effect of
axial compression on the deflections).
Calculate the composite moment of inertia, Ix, for positive moment (deck
in compression) if the girder is to be used for a bridge with 11-foot 6-inchgirder spacing, 4,000 psi deck concrete, 8¼-inch-thick deck, and 2-inch
haunch. The final girder strength is 7,000 psi. Ignore the strand area.
E1.4.
A welded steel plate girder consists of a 75-inch × ½-inch web and
20-inch × 1½-inch-thick flanges. All plates are made from Grade 50W steel.
The girder spacing is 10 ft and the concrete deck is 8¼ inches thick. The
distance from the top of the web to the bottom of the deck is 3½ inches.
Deck concrete strength is 3,000 psi. The modular ratio, n, for 3,000 psi concrete is taken to be equal to 9. Determine the following properties (moment
of inertia, section moduli, and shear flow) for the positive moment condition
(compression in the top):
• Ix, Sxt, Sxb, Q/Ix for the girder top and bottom (girder alone)
• Ix, Sxt, Sxb, Q/Ix for the girder top and bottom (short-term composite)
• Ix, Sxt, Sxb, Q/Ix for the girder top and bottom (long-term composite)
• the web depth in compression in the elastic condition, Dc
• the web depth in compression in the plastic condition, Dcp
• the distance from the top of the deck to the plastic neutral axis, Dp
• the plastic moment of the composite section, Mp
21
22
LRFD Bridge Design
E1.5.
For negative moments (tension in the top), determine the composite section
properties for the girder in problem E1.4. if the area of steel in the deck is
10.00 in2 with f y = 60 ksi, located 3½ inches from the top of the deck.
E1.6.
A welded steel plate girder consists of a 42-inch × ½-inch web and
16-inch × 1-inch thick flanges. All plates are made from Grade 50W steel.
The girder spacing is 10 ft and the concrete deck is 8¼ inches thick. The
distance from the top of the web to the bottom of the deck is 3½ inches.
Deck concrete strength is 4,000 psi. The modular ratio, n, for 4,000 psi
concrete is taken to be equal to 8. Determine the distance from the top of
the deck to the plastic neutral axis (PNA) and the composite plastic moment
in positive bending, Mp.
E1.7.
Estimate the total superstructure weight (kips per foot) for the Project
Bridge. For the concrete girder option, use BT-54 girders. For the steel
girder option, use W40 × 215 rolled steel beams. Deck thickness is 8.25
inches. Haunch thickness is 1.75 inches. Each parapet requires 0.0926 cubic
yards of concrete per linear foot.
E1.8.
For the concrete girder option of the Project Bridge, estimate the total vertical reaction at the intermediate pier due to all dead loads.
2
Loads on Bridges
Bridge loads may be broadly categorized as permanent or transient, with transient
loads further divided into three sub-categories: those resulting from traffic, those
resulting from environmental sources, and those due to extreme events. Some of
the more routinely encountered loads in each category are defined below, based on
Chapter 3 of the Load and Resistance Factor Design-Bridge Design Specifications
(LRFD-BDS) (AASHTO, 2020).
A. Permanent Loads
• DC is the dead load of all structural components, as well as any nonstructural attachments.
• DW is the dead load of additional nonintegral wearing surfaces, future
overlays, and any utilities supported by the bridge.
• EV is the vertical earth pressure from the dead load of earth fill.
• EH is the load due to horizontal earth pressure.
• DD are the loads developed along the vertical sides of a deep-foundation
element tending to drag it downward, typically due to consolidation of
soft soils underneath embankments reducing its resistance.
B. Transient Loads – traffic
• LL is the vertical gravity load due to vehicular traffic.
• PL represents the vertical gravity load due to pedestrian traffic.
• IM represents the dynamic load allowance to amplify LL.
• BR is the horizontal vehicular braking force.
• CE is the horizontal centrifugal force from vehicles on a curved
roadway.
C. Transient Loads – environment
• WA is the pressure due to differential water levels, stream flow, or
buoyancy.
• WS is the horizontal and vertical pressure due to wind.
• WL is the horizontal pressure on vehicles due to wind.
• TU is the uniform temperature change.
• TG is the temperature gradient.
• SE is the effect of settlement.
• FR represents the frictional forces on sliding surfaces.
D. Transient Loads – extreme event
• BL represents the intentional or unintentional forces due to blasting.
• EQ represents loads due to earthquake ground motions.
• CT represents horizontal impact loads due to vehicles or trains.
• CV represents horizontal impact loads due to aberrant ships or barges.
• IC is the horizontal static and dynamic force due to ice action.
• SE is the effect of settlement.
DOI: 10.1201/9781003265467-2
23
24
LRFD Bridge Design
The basic load and resistance factor design (LRFD) relationship is expressed in
Equation 2.1, with the load modifier given by Equation 2.2. The load modifier, ηi,
consists of three components to account for ductility, redundancy, and importancebased considerations.
Qn = åhig iQi £ f Rn
(2.1)
hi = h Dh Rh I
(2.2)
The load modifier, ηi, is to be taken to be no less than 0.95 for cases in which the use
of maximum load factors, γi, are used. For cases in which the use of minimum load
factors, γi, are used, ηi shall be taken no greater than 1.0.
The ductility-related modifier, ηD, is equal to 1.00 for all limit states except the
Strength limit states. For the Strength limit state, ηD is equal to:
• 1.00 for conventional designs
• 1.05 for non-ductile components and connections
• 0.95 (or more) for elements designed using enhanced ductility measures
beyond those required by the LRFD-BDS
The redundancy-related modifier, ηR, is equal to 1.00 for all limit states except the
Strength limit states. For the Strength limit state, ηR is equal to:
• 1.00 for conventional designs
• 1.05 for non-redundant members
• 0.95 (or more) for exceptional levels of redundancy
The operational importance-related modifier, ηI, is equal to 1.00 for all limit states
except the Strength limit states. For the Strength limit state, ηI is equal to:
• 1.00 for typical bridges
• 1.05 for critical or essential bridges
• 0.95 (or more) for bridges deemed to be relatively less important
The following sections provide a discussion of some of the most frequently encountered loads on bridges. See Chapter 3 for a discussion on limit states for bridges.
2.1 DEAD LOADS (DC AND DW)
DC loads for a typical bridge include the girder self-weight and associated framing (cross-frames, diaphragms, lateral bracing), the weight of the concrete deck and
haunch (also known as filler; the area between the bottom of the deck and the top of
the girder, filled with deck concrete but often ignored in section property calculations), sidewalks, if present, and parapets.
25
Loads on Bridges
DW loads include an allowance for future overlay, typically about 35 psf (pounds
per square foot). Also included in the DW loading is the weight of any utilities
attached to the structure.
Unit weights for DC load computation are typically taken to be equal to 150 pcf
(pounds per cubic foot) for concrete and 490 pcf for steel.
2.2 LIVE AND IMPACT LOADS (LL AND IM)
The traffic live load in the American Association of State Highway and Transportation
Officials (AASHTO) LRFD-BDS is the HL-93 live load, consisting of a design truck
with a simultaneously applied uniform design lane load. The effect of a design tandem with the simultaneously applied uniform design lane load is also to be included.
The design lane load is 0.640 klf (kips per linear foot). The design tandem consists of two 25-kip axles 5 feet apart. The design truck is a 72-kip truck with variable
rear axle spacing, as depicted in Figure 2.1.
For the Fatigue limit state, a single truck is placed in a single lane. The Fatigue
truck is different from the design truck only in the rear axle spacing, which is constant at 30 feet for the Fatigue truck.
For limit states other than Fatigue, multiple loaded lanes must be considered in
the design of the bridge elements. To account for the rarity of maximum loading
placement in multiple lanes simultaneously, a multi-presence factor, m, is applied to
analytical results. This factor varies, depending on the number of lanes loaded, as
shown in Table 2.1.
FIGURE 2.1
HL-93 design truck.
TABLE 2.1
Multi-presence factor m
Number of Lanes Loaded
Multi-presence factor, m
1
1.20
2
3
1.00
0.85
4 or more
0.65
26
LRFD Bridge Design
Even though the Fatigue load, as previously mentioned, is a single truck in a
single lane, the multi-presence factor, m, is not to be applied to the Fatigue limit state
results. The multi-presence factor does apply for all other limit states.
Since traffic loads are not static, an impact factor, IM, is incorporated into the
live load effects for the truck load only. No impact factor is included in the uniform
lane loading. The LL+IM effect is the truck portion of the LL effect multiplied by
(1+IM). IM is equal to:
• 0.15 for the Fatigue limit state
• 0.33 for all other limit states
• 0.75 for the design of deck joints
Bridges are designed for the number of full lanes which will physically fit on the
available width. A lane is assumed to occupy a 12-foot width. Partial lanes need not
be considered. For example, the Project Bridge will need to be designed for a maximum of two lanes since the available width is 32 feet (the integer portion of 32/12
is 2). Some components will be controlled by the condition for which a single lane
is loaded, and other components will be controlled by the condition for which two
lanes are loaded. The deck overhang, for example, will be controlled by the singlelane case, with m = 1.2, given that vehicles inside the exterior girder contribute no
added effect to overhang deck moment. Maximum vertical load on the intermediate
pier will be controlled by the two-lane case, with m = 1.0.
The uniform lane load and the design truck, with six feet between the wheels on
a given axle, are assumed to occupy a 10-foot width and are moved laterally within
the 12-foot wide lane. The design truck exterior wheel is to be placed no closer than
1.0 foot from the edge of the lane for deck design, and no closer than 2.0 feet from
the edge of the lane for all other analyses.
As previously mentioned, the design live loading for bridges is one design truck plus
the uniform lane load of 0.640 klf in each loaded lane. However, for girder reactions at
intermediate, continuous supports, and for negative girder moments between points of
contraflexure, the AASHTO LRFD-BDS requires that 90% of the effect of two trucks
plus 90% of the effect of the lane load in each loaded lane be considered as well.
To summarize the application of the HL-93 live load on bridges, the extreme
effect is the larger of the following:
a) the effect of the design tandem combined with the effect of the design lane
load, or
b) the effect of one design truck with the variable axle spacing combined with
the effect of the design lane load, or
c) for negative moment between points of contraflexure and for reactions at
interior piers only, 90 percent of the effect of two design trucks combined
with 90 percent of the effect of the design lane load.
For load placement in (c), the trucks are to be placed with a minimum of 50.0 ft
between the lead axle of the second truck and the rear axle of the first truck, and the
27
Loads on Bridges
distance between the rear 32.0-kip axles of each truck is to be 14.0 ft. Since case (c)
applies only for negative moment and pier reactions, the two design trucks are to be
placed in adjacent spans. It would thus seem that it is seldom, if ever, necessary to
place two trucks in the same span for the HL-93 live loading.
2.3 BRAKING FORCES (BR)
Braking forces occur when traffic slows on a bridge deck. The design braking force
is taken to be applied 6 feet above the surface of the deck, is placed in all lanes carrying traffic in the same direction, and is equal to the larger of:
a) 25 percent of the axle weights of the design truck or design tandem, or
b) 5 percent of the design truck plus lane load, or
c) 5 percent of the design tandem plus lane load
Braking forces should be distributed to substructures (abutments and piers) according to the relative stiffness of each substructure.
With expansion joints at each abutment, a typical assumption would be that the
piers (bents) carry 100% of the braking forces. With integral abutments, the abutments are likely to carry a large proportion of the braking forces. In such cases, a
lower bound abutment stiffness might be useful in estimating the fraction of braking
forces to be assigned to the piers, with an upper bound abutment stiffness used to
determine braking forces assigned to the abutments.
2.4 CENTRIFUGAL FORCES (CE)
Centrifugal forces occur when traffic moves on a bridge which is curved in plan.
With effects from opposing lanes on a two-directional, two-lane bridge offsetting
one another, one lane of centrifugal force (with m = 1.2) may suffice. However, it may
also be necessary to consider the possibility of a bridge becoming one-directional in
the future and designing for such a condition with multiple lanes of centrifugal force,
and an appropriate multi-presence factor.
The centrifugal force (FCE ) imparted to a bridge is a horizontal force to be applied
transversely to the direction of travel (away from the center of curve), six feet above
the deck, and is given by Equation 2.3.
FCE = CW
(2.3)
v2
gR
(2.4)
C= f
where W is the weight of the truck (72 kips) or tandem (50 kips). The design speed,
v, must be expressed in units of ft/s, g is the acceleration of gravity (32.2 ft/s2), and
R is the radius of curvature of the loaded lane (ft). The factor, f, is 1.0 for the Fatigue
limit state and 4/3 for all other limit states.
28
LRFD Bridge Design
Centrifugal forces alter the weight distribution on the design truck wheels. For a
stationary vehicle, 50% of the weight goes to each wheel line. It may be shown that
the distribution of vehicle weight to the wheel lines is given by Equations 2.5 through
2.10, with reference to Figure 2.2 for the definition of the various force components.
The effect can be significant and, although often ignored in the computation of live
load distribution factors, may need to be considered for curved structures.
FN 2
W
éæ h ö
ù éæ 1 ö
ù
æ1ö
æhö
= C êç ÷ cos f + ç ÷ sin f ú + êç ÷ cos f - ç ÷ sin f ú
è2ø
èbø
ëè b ø
û ëè 2 ø
û
FN 1
W
(
= cos f + C ( sin f ) - FN 2
FT 1
FT 2
FV 1
W
(
=
W
W
W
)
C ( cos f ) - sin f
1 + FN 2
FN 1
(
= FT 1
= FN 1
FIGURE 2.2 Centrifugal forces.
(
W
)
W
)×(F
N2
FN 1
(2.6)
(2.7)
)
) cosf - ( F W ) (sin f )
T1
(2.5)
(2.8)
(2.9)
29
Loads on Bridges
FV 1
W
(
= FN 1
W
) cosf - ( F W ) (sin f )
T1
(2.10)
FV1/W and FV2/W are the relative vertical components of load distributed to each
wheel line and may be useful in the lever rule or in the rigid cross-section methods
for live load distribution factor calculations (see Chapter 5). Note that, for the design
truck, h = b = 6 feet.
Suppose R = 730 ft and the superelevation, SE = 0.08, with the design speed equal
to 50 mph. Solving the equations, it may be advisable in such a case to use a wheel
load distribution of 0.28 (inner) and 0.72 (outer), rather than the usual 0.50 (inner)
and 0.50 (outer), for live load distribution.
Note as well that the typical coefficient of friction, μ, between dry pavement and
average tires is 0.80, between wet pavement and a grooved tire is 0.70, and between
wet pavement and a smooth tire is 0.40. In all three cases, the friction is sufficient to
resist the tangential forces since FT1/FN1 = FT2/FN2 = 0.220. This means that a vehicle
travelling at the 50-mph design speed, with the specified R = 730 ft and SE = 0.08, is
unlikely to slide.
2.5 WIND LOADS (WS AND WL)
Wind loads in the AASHTO LRFD-BDS are based on 3-second gust wind speeds.
This was not always the case, as previous versions used fastest-mile wind speed as
the design basis. It is important not to mix codes and specifications in structural
design for this, and other, reasons. Load factors and resistance factors vary among
design standards.
For the Strength III limit state, wind speeds are taken from maps in the AASHTO
LRFD-BDS. For the Service IV limit state, the wind speed is taken as 75% of that
for the Strength III limit state. Other limit states for which wind loading is applied
include Service I, for which a wind speed of 70 mph is used, and Strength V, for
which the design wind speed is 80 mph. See Chapter 3 for a detailed discussion of
limit states.
The basic horizontal wind pressure equation is given here in Equation 2.11. The
velocity, V, must be expressed in mph (miles per hour). The resulting pressure is
in ksf (kips per square foot). The gust factor, G, is 1.0 for structures other than
sound barriers. The drag coefficient, CD, is 1.3 for I-girder superstructures and 1.6 for
bridge substructures. The surface exposure and elevation coefficient, K Z, accounts
for height above ground (or water) and surface roughness conditions. Maps in the
AASHTO LRFD-BDS or Equations 2.12 through 2.14 may be used to determine KZ
for Strength III and Service IV limit states. For all other limit states, K Z is equal to
1.0. The height above ground or water, Z, is never to be taken less than 33 feet. Wind
loads on substructures are to be computed using the superstructure height above
ground or water, unless otherwise approved by the Owner.
PZ = 2.56 ´ 10 -6 ´ V 2 K Z GC D
(2.11)
30
LRFD Bridge Design
é
ù
æ Z ö
ê2.5 ln ç 0.9834 ÷ + 6.87 ú
è
ø
û
KZ ( B) = ë
345.6
2
é
ù
æ Z ö
ê2.5 ln ç 0.0984 ÷ + 7.35ú
è
ø
û
K Z (C ) = ë
478.4
2
é
ù
æ Z ö
ê2.5 ln ç 0.0164 ÷ + 7.65ú
è
ø
û
KZ ( D ) = ë
616.1
2
(2.12)
(2.13)
(2.14)
Ground surface roughness ‘B’ generally refers to “terrain with numerous
closely spaced obstructions having the size of single-family dwellings or larger”
(AASHTO, 2020).
Ground surface roughness ‘C’ generally refers to “open terrain with scattered
obstructions having heights generally less than 33.0 ft, including flat open country
and grasslands” (AASHTO, 2020).
Ground surface roughness ‘D’ generally refers to “flat, unobstructed areas
and water surfaces”, including “smooth mud flats, salt flats, and unbroken ice”
(AASHTO, 2020).
The reader is referred to Section 3.8 of the LRFD-BDS for detailed discussions on
the determination of appropriate ground surface roughness designations.
Multiple angles of wind direction are typically investigated. For the case in which
the wind is perpendicular to the bridge (and only for this case), a vertical (upward)
pressure of 20 psf (Strength III) or 10 psf (Service IV) is applied to the bridge deck
at one-quarter of the distance from the windward edge of the bridge. This load acts
concurrently with the horizontal wind pressures and is not applied for limit states
other than Strength III and Service IV.
In addition to wind load on the structure, WS, AASHTO also requires consideration of wind load on the live load, WL. The wind load on traffic is taken to be
0.10 klf transverse to traffic and 0.04 klf in the direction of traffic, both applied 6
feet above the deck surface.
2.6 COLLISION LOADS (CT AND CV)
For substructures located closer than 30 feet (typical clear zone) to the roadway
edge, and with no independent barrier designed to withstand vehicle collision,
the substructure components must be designed to withstand such collision loads.
Vehicle collision loading (CT) is defined as a 600-kip load, five feet above ground
at an angle of between 0 and 15 degrees with the pavement edge. This vehicle
31
Loads on Bridges
collision load is based on crash tests of rigid columns subjected to an 80-kip vehicle
at a speed of 50 mph.
As an alternative to designing substructures within 30 feet of the roadway for the
vehicular collision load, an independent barrier protecting the substructure may be
used. Such barriers may be any of the following:
• An embankment;
• A structurally independent, crashworthy, ground-mounted 54-inch-high
barrier, located within 10 ft of the component being protected;
• A 42-inch high barrier located at more than 10 ft from the component being
protected.
Such barriers are required to be structurally and geometrically capable of surviving
the crash test for Test Level 5 as specified in Section 13 of the AASHTO LRFD-BDS.
Bridge parapets and decks are designed for vehicular crash load as well.
Table 2.2 summarizes the various test levels specified in Section 13 of the
AASHTO-LRFD-BDS.
The parameters are defined as follows:
•
•
•
•
•
•
Ft = transverse impact force applied at height He, kips
Fl = longitudinal impact force applied at height He, kips
Fv = vertical impact force applied at height He, kips
H = height of wall, ft
Lt = longitudinal length of distribution of horizontal impact force, ft
Lv = longitudinal length of distribution of vertical impact force, ft
Bridge components in navigable waterways must be designed for vessel collision
(CV) loading or protected by barriers such as dolphins, fenders, or berms. The interested reader is referred to Section 3.14 of the AASHTO-LRFD-BDS for guidance on
vessel collision load determination.
TABLE 2.2
Crash Load Parameters for Various Test Levels
Parameter
TL-1
TL-2
TL-3
TL-4
TL-5
TL-6
Ft, transverse (kips)
13.5
27.0
54.0
54.0
124.0
175.0
FL, longitudinal (kips)
Fv, vertical (kips)
Lt and LL (feet)
Lv (feet)
He (min, inches)
4.5
4.5
4.0
18.0
18.0
9.0
4.5
4.0
18.0
20.0
18.0
4.5
4.0
18.0
24.0
18.0
18.0
3.5
18.0
32.0
41.0
80.0
8.0
40.0
42.0
58.0
80.0
8.0
40.0
56.0
Min rail height, H (ft)
27.0
27.0
27.0
32.0
42.0
90.0
32
LRFD Bridge Design
2.7 TEMPERATURE LOADS (TU)
For thermal expansion and contraction requirements, sites are classified as either
‘moderate climate’ or ‘cold climate’ conditions in the AASHTO-BDS. The reader is
referred to Section 3.12 in the AASHTO-BDS for maps.
Figure 2.3 depicts a scenario for a five-span bridge with piers of varying height.
Although some engineers assume the ‘center of stiffness’ to be midway between the
bridge ends, such an assumption can result in serious error. For each substructure
element contributing to the longitudinal stiffness of the system, Equation 2.15 may
be used to establish the location of the center of stiffness – the stationary point on the
superstructure from which expansion and contraction due to temperature variation
take place. Although these estimates are approximate due to the difficulty in estimating substructure stiffness (Ki) values (particularly for abutments without expansion
joints; for abutments with expansion joints, the abutment stiffness may be taken to
be equal to zero), they are often preferable to traditional assumptions. The center
of stiffness is located L A1 from abutment number 1. The distance from abutment 1
to each substructure is xi. Equation 2.16 gives the total thermal movement required
at a given substructure location. As shown in Figure 2.3, Li is the distance from the
center of stiffness to the substructure in question.
L A1 =
FIGURE 2.3
å K i × xi
å Ki
Thermal expansion calculation schematic.
(2.15)
33
Loads on Bridges
( DTU )i = a (d T ) Li
(2.16)
For bridges located in moderate climates, the design temperature range, δT is as
follows:
• δT = 0°–120° for steel girder superstructures
• δT = 10°–80° for concrete girder superstructures
For bridges located in cold climates, the design temperature range, δT is as follows:
• δT = −30°–+120° for steel girder superstructures
• δT = 0°–80° for concrete girder superstructures
The coefficient of thermal expansion, α, is 0.0000060/°F for concrete girder superstructures and 0.0000065/°F for steel girder superstructures.
Equation 2.16 gives the total theoretical movement, which may then be broken
into expansion and contraction components. In an idealized situation, one-half of the
total movement would be expansion and one-half would be contraction. To account
for variations in temperature at the time of structure completion and inaccuracies
inherent in such computations, the AASHTO LRFD-BDS adopts a load factor of 1.2
on TU deformation loads at both the Service and Strength limit states (see Chapter 3,
Table 3.1 of this book). Therefore, the design total movement may be taken as 1.2
times that given by Equation 2.16, resulting in expansion and contraction requirements each equal to 60% of that given by Equation 2.16. Engineers may wish to add
an additional contingency to calculated movements, given the uncertainties involved
in the estimates. The method has been used successfully on many projects. Note,
however, that for the design of reinforced elastomeric bearings, Section 14.7.5.3.2 of
the LRFD-BDS requires that 65% (rather than 60%) of the total movement be used
in the design of such bearings.
2.8 EARTHQUAKE LOADS (EQ)
Seismic loading in the AASHTO LRFD-BDS is defined as a geometric-meanbased, uniform hazard, response spectrum. This is in contrast to building design in
accordance with ASCE 7-16, which defines earthquake ground motion for design as
the maximum-direction-based, risk-targeted, response spectrum. Such definitions
in design codes and specifications change frequently, and engineers would be well
served by keeping abreast of changes in the nature of seismic load definitions in
AASHTO, in ASCE 7, and in any applicable design standard.
A Poisson probability distribution is typically used as the model defining relationships among exposure time, t, probability of exceedance, PE, and mean recurrence
interval, MRI. The model is given here in Equation 2.17.
MRI =
-t
ln (1 - PE )
(2.17)
34
LRFD Bridge Design
The shape of the design response spectrum is depicted in Figure 2.4. Determination
of the control points (AS, SDS, SD1, TS, TO) on the design response spectrum requires
the determination of an appropriate site class for the project. A response spectrum is
a plot of pseudo-spectral acceleration, PSA, versus structure period, T.
The code-based shape is entirely determined by the control points. However, current trends in building design may result in a 22-point design response spectrum with
site effects incorporated automatically. This provided further impetus for engineers
to be fully aware of the nature and basis of design response spectra in the governing,
current design specification.
As of December 2020, the design ground motion in AASHTO is that having a
7% probability of exceedance in 75 years. With t = 75 years and PE = 0.07, Equation
2.17 gives MRI = 1,033 years. The USGS Unified Hazard Tool will be useful in determining uniform hazard, geometric-mean-based PSA values for a site with a given
latitude and longitude. The application URL is listed below.
https://earthquake​.usgs​.gov​/ hazards​/interactive/
Site class determination is based on either average shear wave velocity in the upper
100 feet of the subsurface profile, or, more frequently, on the average standard penetration test blow count in the upper 100 feet. The shear wave velocity averaged
over the upper 100 feet (30 meters) of the soil profile is termed VS30. It is important
FIGURE 2.4 AASHTO design response spectrum control points.
35
Loads on Bridges
to recognize that a velocity-based average is not computed in the same way as a
distance-based average is calculated. The correct calculation of VS30 is given in
Equations 2.18 and 2.19. Only the top 30 meters of the profile is used to calculate
VS30. Two calculations are necessary: (1) consider all layers in the top 30 meters and
(2) consider only the cohesionless layers in the top 30 meters.
(VS 30 )1 =
(VS 30 )2 =
å di
, all layers
di
å
VSi
(2.18)
å di
, include cohesionless layers ONLY
d
å i
VSi
(2.19)
Whereas a true average blow count would be computed differently, the same format
is to be used when blow count data is selected as the basis for site classification, with
VSi replaced by Ni (blow count) in Equations 2.18 and 2.19. Ni is never to be taken
larger than 100 blows per foot and should generally be taken as equal to 100 blows
per foot for rock, when such occurs at a depth of less than 100 feet.
Table 2.3 summarizes criteria used to establish the appropriate site class for a
project. In Table 2.3, Su is the undrained shear strength.
The long transition period, TL, is not available in the USGS Unified Hazard Tool
but is available in the ATC Hazard by Location online Tool. The ATC application
produces design PSA control points for buildings and should not be used to determine parameters other than TL for bridges at this time. The tool URL is shown below.
https://hazards​.atcouncil​.org/
Site factors are to be applied to PSA values at the B/C boundary. Note that the codebased site factors are typically 1.00 for Site Class B in current codes and specifications for bridges. PGA, SS, and S1 are mapped accelerations at the B/C boundary. As
Figure 2.4 shows, these three pseudo-spectral accelerations are multiplied by site
factors, FPGA, Fa, and Fv, respectively, to define the design response spectrum.
TABLE 2.3
Site Class Definitions
Site Class
A. Hard Rock
B. Rock
C. Very Dense Soil/Soft Rock
D. Stiff Soil
E. Soft Clay Soil
F. Soils Requiring Site Response
VS30, ft/sec
N, blows/ft
>5,000
NA
Su, psf
NA
2,500–5,000
1,200–2,500
600–1,200
<600
NA
>50
15–50
<15
NA
>2,000
1,000–2,000
<1,000
Liquefaction, peats, highly sensitive or plastic clays
36
LRFD Bridge Design
Code-based site factors from the AASHTO LRFD-BDS are summarized in
Tables 2.4 through ​2.6. However, these site factors may not be suitable for areas
with deep soil profiles, such as are found in the Mississippi Embayment (ME) of
the New Madrid seismic zone (NMSZ), among other regions. Research performed
in the past 15 years or so has produced site factors for deep soil sites much different
from code-based values. Tables 2.7 and 2.8 summarize such findings from one such
study (Hashash et al., 2008). Tables 2.9 through ​​2.12 summarize results from another
(Malekmohammadi and Pezeshk, 2014). Both studies distinguish between so-called
“uplands” and “lowlands” sites within the ME.
Engineers involved in structural design on deep soil sites may wish to consider
alternative site factors, such as those presented here.
The multitude of seismic design spectrum options is evident through browsing the
USGS Seismic Design Web Services page:
https://earthquake​.usgs​.gov​/ws​/designmaps/
TABLE 2.4
AASHTO Fpga Site Coefficient
Fpga
PGA Range of Applicability
Site Class
0.10
0.20
0.30
0.40
0.50
A
0.80
0.80
0.80
0.80
0.80
B
C
D
1.00
1.20
1.60
1.00
1.20
1.40
1.00
1.10
1.20
1.00
1.00
1.10
1.00
1.00
1.00
E
2.50
1.70
1.20
0.90
0.90
TABLE 2.5
AASHTO Fa Site Coefficient
Fa
SS Range of Applicability
Site Class
0.25
0.50
0.75
1.00
1.25
A
0.80
0.80
0.80
0.80
0.80
B
C
D
1.00
1.20
1.60
1.00
1.20
1.40
1.00
1.10
1.20
1.00
1.00
1.10
1.00
1.00
1.00
E
2.50
1.70
1.20
0.90
0.90
37
Loads on Bridges
TABLE 2.6
AASHTO Fv Site Coefficient
Fv
S1 Range of Applicability
Site Class
0.10
0.20
0.30
0.40
0.50
A
0.80
0.80
0.80
0.80
0.80
B
C
D
1.00
1.70
2.40
1.00
1.60
2.00
1.00
1.50
1.80
1.00
1.40
1.60
1.00
1.30
1.50
E
3.50
3.20
2.80
2.40
2.40
TABLE 2.7
Site Factor Fa from Hashash et al. (2008)
SS = 0.25
Thickness (meters)
Up
Low
SS = 0.50
Up
Low
SS = 0.75
Up
Low
SS = 1.00
Up
Low
SS ≥ 1.25
Up
Low
30
1.46
1.41
1.32
1.27
1.18
1.13
1.11
1.06
1.06
1.01
100
200
300
500
1.41
1.36
1.31
1.27
1.31
1.21
1.11
1.06
1.27
1.22
1.17
1.13
1.17
1.07
0.97
0.92
1.13
1.08
1.03
0.99
1.03
0.93
0.83
0.78
1.06
1.01
0.96
0.92
0.96
0.86
0.76
0.71
1.01
0.96
0.91
0.87
0.91
0.81
0.71
0.66
1000
1.23
1.04
1.09
0.90
0.95
0.76
0.88
0.70
0.83
0.64
‘Up’ = Uplands; ‘Low’ = Lowlands.
TABLE 2.8
Site Factor Fv from Hashash et al. (2008)
S1 = 0.10
S1 = 0.20
S1 = 0.30
S1 = 0.40
S1 ≥ 0.50
Thickness (meters)
Up
Low
Up
Low
Up
Low
Up
Low
Up
30
2.40
2.40
2.00
2.00
1.80
1.80
1.60
1.60
1.50
1.50
100
200
300
500
2.70
2.85
2.95
3.00
2.55
2.67
2.77
2.82
2.30
2.45
2.55
2.60
2.15
2.27
2.37
2.42
2.10
2.25
2.37
2.42
1.95
2.07
2.17
2.22
1.95
2.08
2.18
2.23
1.75
1.87
1.97
2.02
1.80
1.91
2.01
2.06
1.65
1.77
1.87
1.92
1000
3.05
2.87
2.65
2.47
2.47
2.27
2.28
2.07
2.08
1.97
‘Up’ = Uplands; ‘Low’ = Lowlands.
Low
38
LRFD Bridge Design
TABLE 2.9
Uplands Site Factor Fa from Malekmohammadi and Pezeshk (2014)
VS30, m/s
560
Site Class
Depth, m
SS ≤ 0.25
SS = 0.50
SS = 0.75
SS = 1.00
SS ≥ 1.25
C
30
1.509
1.228
1.049
0.923
0.829
70
140
400
750
30
70
140
400
750
30
70
140
400
1.624
1.618
1.362
1.069
1.528
1.660
1.667
1.421
1.123
1.451
1.581
1.592
1.362
1.285
1.250
1.011
0.774
1.057
1.117
1.095
0.896
0.691
0.900
0.954
0.938
0.771
1.081
1.036
0.819
0.617
0.803
0.836
0.807
0.646
0.491
0.638
0.666
0.645
0.518
0.940
0.892
0.693
0.517
0.647
0.666
0.637
0.501
0.377
0.490
0.505
0.485
0.383
0.837
0.788
0.604
0.447
0.543
0.553
0.525
0.408
0.304
0.396
0.405
0.385
0.301
750
1.079
0.595
0.394
0.288
0.225
270
D
180
E
TABLE 2.10
Uplands Site Factor Fv from Malekmohammadi and Pezeshk (2014)
VS30, m/s
560
Site Class
C
270
D
180
E
Depth, m
S1 ≤ 0.10
S1 = 0.20
S1 = 0.30
S1 = 0.40
S1 ≥ 0.50
30
3.304
2.841
2.550
2.340
2.179
70
140
400
750
30
70
140
400
750
30
70
140
400
4.428
5.630
4.171
3.559
3.771
4.383
4.974
4.397
4.170
3.604
4.007
4.390
4.099
3.862
4.947
3.708
3.181
2.753
3.245
3.711
3.318
3.164
2.341
2.640
2.914
2.753
3.496
4.498
3.394
2.921
2.176
2.586
2.970
2.674
2.558
1.703
1.937
2.148
2.042
3.227
4.165
3.158
2.724
1.803
2.155
2.483
2.246
2.153
1.327
1.518
1.688
1.613
3.017
3.904
2.971
2.567
1.543
1.853
2.140
1.943
1.866
1.083
1.244
1.387
1.330
750
4.017
2.713
2.019
1.598
1.320
39
Loads on Bridges
TABLE 2.11
Lowlands Site Factor Fa from Malekmohammadi and Pezeshk (2014)
VS30, m/s
560
Site Class
Depth, m
SS ≤ 0.25
SS = 0.50
SS = 0.75
SS = 1.00
SS ≥ 1.25
C
30
2.156
1.844
1.622
1.457
1.330
70
140
400
750
30
70
140
400
750
30
70
140
400
2.159
2.068
1.668
1.283
2.182
2.207
2.131
1.740
1.348
2.072
2.103
2.035
1.668
1.780
1.653
1.273
0.953
1.586
1.546
1.448
1.129
0.851
1.351
1.321
1.241
0.971
1.532
1.398
1.048
0.772
1.242
1.185
1.090
0.827
0.614
0.987
0.944
0.871
0.663
1.355
1.221
0.898
0.654
1.022
0.960
0.872
0.649
0.476
0.773
0.729
0.663
0.496
1.222
1.091
0.790
0.570
0.870
0.808
0.727
0.533
0.388
0.636
0.592
0.534
0.393
750
1.296
0.743
0.493
0.365
0.286
270
D
180
E
TABLE 2.12
Lowlands Site Factor Fv from Malekmohammadi and Pezeshk (2014)
VS30, m/s
560
Site Class
C
270
D
180
E
Depth, m
S1 ≤ 0.10
S1 = 0.20
S1 = 0.30
S1 = 0.40
S1 ≥ 0.50
30
3.366
3.131
2.944
2.792
2.666
70
140
400
750
30
70
140
400
750
30
70
140
400
4.494
5.702
4.215
3.593
3.842
4.449
5.038
4.443
4.210
3.671
4.067
4.446
4.142
4.175
5.290
3.918
3.344
3.035
3.509
3.968
3.507
3.326
2.581
2.854
3.116
2.909
3.921
4.965
3.682
3.144
2.512
2.901
3.278
2.901
2.753
1.966
2.173
2.370
2.216
3.716
4.702
3.490
2.982
2.150
2.482
2.803
2.482
2.358
1.583
1.748
1.906
1.783
3.546
4.486
3.332
2.848
1.888
2.178
2.459
2.179
2.070
1.325
1.462
1.594
1.492
750
4.055
2.852
2.173
1.749
1.464
40
LRFD Bridge Design
It is important for engineers to know the basis (required MRI) and nature (risktargeted or uniform hazard, geomean or maximum direction, three-point or twentytwo-point, etc.) of the required design response spectra, and to use the appropriate
tools (site factors if required, web-based applications, etc.) to generate such spectra.
A third, more recent, alternative for site factors in the New Madrid seismic zone
has been summarized in the Korean Society of Civil Engineers Journal (Moon et al.,
2016). Tables 2.13 through ​​​​2.18 summarize the recommended site factors. The developed site factors are compared to National Earthquake Hazards Reduction Program
(NEHRP) site factors in the report and in the Tables.
Given the multitude of uncertainties in site response analysis and in geotechnical investigations to determine soil properties, it may be more prudent to select the
TABLE 2.13
Mississippi Embayment Site Factor Fa from Moon et al. (2016) –Site Class C
SS
NEHRP
Upland
Lowland
30
H, m
0.25
1.20
1.80
1.72
30
30
30
30
100
100
100
100
100
200
200
200
200
200
300
300
300
300
300
500
500
500
500
500
1,000
1,000
1,000
1,000
0.50
0.75
1.00
1.25
0.25
0.50
0.75
1.00
1.25
0.25
0.50
0.75
1.00
1.25
0.25
0.50
0.75
1.00
1.25
0.25
0.50
0.75
1.00
1.25
0.25
0.50
0.75
1.00
1.20
1.10
1.00
1.00
1.20
1.20
1.10
1.00
1.00
1.20
1.20
1.10
1.00
1.00
1.20
1.20
1.10
1.00
1.00
1.20
1.20
1.10
1.00
1.00
1.20
1.20
1.10
1.00
1.75
1.66
1.57
1.54
1.73
1.68
1.59
1.51
1.48
1.67
1.61
1.53
1.46
1.42
1.62
1.56
1.48
1.40
1.37
1.54
1.49
1.40
1.31
1.28
1.39
1.32
1.24
1.14
1.68
1.62
1.49
1.47
1.69
1.65
1.57
1.46
1.43
1.63
1.59
1.51
1.40
1.37
1.58
1.53
1.45
1.34
1.31
1.52
1.46
1.35
1.24
1.21
1.38
1.29
1.17
1.07
1,000
1.25
1.00
1.11
1.05
41
Loads on Bridges
TABLE 2.14
Mississippi Embayment Site Factor Fa from Moon et al. (2016) – Site Class D
SS
NEHRP
Upland
Lowland
30
H, m
0.25
1.60
1.65
1.45
30
30
30
30
100
100
100
100
100
200
200
200
200
200
300
300
300
300
300
500
500
500
500
500
1,000
1,000
1,000
1,000
0.50
0.75
1.00
1.25
0.25
0.50
0.75
1.00
1.25
0.25
0.50
0.75
1.00
1.25
0.25
0.50
0.75
1.00
1.25
0.25
0.50
0.75
1.00
1.25
0.25
0.50
0.75
1.00
1.40
1.20
1.10
1.00
1.60
1.40
1.20
1.10
1.00
1.60
1.40
1.20
1.10
1.00
1.60
1.40
1.20
1.10
1.00
1.60
1.40
1.20
1.10
1.00
1.60
1.40
1.20
1.10
1.48
1.21
1.06
0.97
1.54
1.42
1.14
1.00
0.91
1.49
1.39
1.12
0.98
0.88
1.48
1.37
1.09
0.96
0.86
1.46
1.33
1.05
0.92
0.83
1.41
1.27
0.97
0.84
1.34
1.07
0.94
0.84
1.35
1.24
0.97
0.84
0.75
1.28
1.19
0.92
0.79
0.69
1.25
1.14
0.87
0.75
0.66
1.20
1.09
0.82
0.70
0.64
1.15
1.04
0.77
0.66
1,000
1.25
1.00
0.80
0.62
worst-case site factors from tabulated values, rather than interpolating. For example, suppose the Moon et al. (2016) site factors are being used at a lowland, Class
E, Mississippi Embayment site with S1 = 0.45 and a profile depth, H = 450 meters.
Applicable tabulated values bounding the problem are:
•
•
•
•
S1 = 0.40, H = 300 meters ⟶ Fv = 2.68
S1 = 0.50, H = 300 meters ⟶ Fv = 2.33
S1 = 0.40, H = 500 meters ⟶ Fv = 2.83
S1 = 0.50, H = 500 meters ⟶ Fv = 2.54
42
LRFD Bridge Design
TABLE 2.15
Mississippi Embayment Site Factor Fa from Moon et al. (2016) – Site Class E
SS
NEHRP
Upland
Lowland
30
H, m
0.25
2.50
1.62
1.46
30
30
30
30
100
100
100
100
100
200
200
200
200
200
300
300
300
300
300
500
500
500
500
500
1,000
1,000
1,000
1,000
0.50
0.75
1.00
1.25
0.25
0.50
0.75
1.00
1.25
0.25
0.50
0.75
1.00
1.25
0.25
0.50
0.75
1.00
1.25
0.25
0.50
0.75
1.00
1.25
0.25
0.50
0.75
1.00
1.70
1.20
0.90
0.90
2.50
1.70
1.20
0.90
0.90
2.50
1.70
1.20
0.90
0.90
2.50
1.70
1.20
0.90
0.90
2.50
1.70
1.20
0.90
0.90
2.50
1.70
1.20
0.90
1.10
0.92
0.79
0.69
1.56
1.05
0.87
0.75
0.66
1.52
1.03
0.87
0.75
0.66
1.48
1.03
0.86
0.75
0.66
1.44
1.00
0.83
0.72
0.64
1.38
0.94
0.79
0.69
0.99
0.87
0.74
0.64
1.44
0.95
0.81
0.69
0.58
1.44
0.95
0.81
0.69
0.58
1.42
0.95
0.81
0.69
0.58
1.40
0.95
0.81
0.68
0.58
1.34
0.91
0.76
0.63
1,000
1.25
0.90
0.62
0.58
Two-way interpolation would yield Fv = 2.64. While this value would likely be permissible, a more logical solution might be to take Fv = 2.83, the largest of the four
tabulated values surrounding the prescribed site conditions.
Based on the PSA at a 1-second period, SD1, a “Seismic Zone” is assigned in the
LRFD-BDS for force-based seismic design.
Based on the PSA at a 1-second period, SD1, a “Seismic Design Category” is
assigned in the LRFD-GS for displacement-based seismic design.
• SD1 ≤ 0.15
• 0.15 < SD1 ≤ 0.30
⟶ Seismic Zone 1 ⟶ Seismic Design Category A
⟶ Seismic Zone 2 ⟶ Seismic Design Category B
43
Loads on Bridges
TABLE 2.16
Mississippi Embayment Site Factor Fv from Moonet al. (2016) – Site Class C
S1
NEHRP
Upland
Lowland
30
H, m
0.10
1.70
1.42
1.40
30
30
30
30
100
100
100
100
100
200
200
200
200
200
300
300
300
300
300
500
500
500
500
500
1,000
1,000
1,000
1,000
0.20
0.30
0.40
0.50
0.10
0.20
0.30
0.40
0.50
0.10
0.20
0.30
0.40
0.50
0.10
0.20
0.30
0.40
0.50
0.10
0.20
0.30
0.40
0.50
0.10
0.20
0.30
0.40
1.60
1.50
1.40
1.30
1.70
1.60
1.50
1.40
1.30
1.70
1.60
1.50
1.40
1.30
1.70
1.60
1.50
1.40
1.30
1.70
1.60
1.50
1.40
1.30
1.70
1.60
1.50
1.40
1.40
1.36
1.35
1.34
1.59
1.53
1.48
1.44
1.41
1.82
1.72
1.61
1.55
1.54
1.98
1.83
1.67
1.55
1.54
1.98
1.83
1.67
1.55
1.54
1.98
1.83
1.67
1.55
1.39
1.38
1.29
1.24
1.76
1.73
1.71
1.55
1.51
2.00
1.91
1.80
1.66
1.65
2.22
2.03
1.81
1.66
1.65
2.22
2.03
1.81
1.66
1.65
2.22
2.03
1.81
1.66
1,000
0.50
1.30
1.54
1.65
• 0.30 < SD1 ≤ 0.50
• SD1 > 0.50
⟶ Seismic Zone 3 ⟶ Seismic Design Category C
⟶ Seismic Zone 4 ⟶ Seismic Design Category D
2.9 WATER LOADING (WA)
Stream flow pressure exerted on piers in the longitudinal direction is determined by
Equation 2.20. It is common practice to align substructure pier columns to coincide
with the direction of flow, in which case the lateral pressure exerted on the pier columns is zero. In cases where the pier axis is not aligned with the direction of flow,
44
LRFD Bridge Design
TABLE 2.17
Mississippi Embayment Site Factor Fv from Moon et al. (2016) – Site Class D
S1
NEHRP
Upland
Lowland
30
H, m
0.10
2.40
2.20
2.30
30
30
30
30
100
100
100
100
100
200
200
200
200
200
300
300
300
300
300
500
500
500
500
500
1,000
1,000
1,000
1,000
0.20
0.30
0.40
0.50
0.10
0.20
0.30
0.40
0.50
0.10
0.20
0.30
0.40
0.50
0.10
0.20
0.30
0.40
0.50
0.10
0.20
0.30
0.40
0.50
0.10
0.20
0.30
0.40
2.00
1.80
1.60
1.50
2.40
2.00
1.80
1.60
1.50
2.40
2.00
1.80
1.60
1.50
2.40
2.00
1.80
1.60
1.50
2.40
2.00
1.80
1.60
1.50
2.40
2.00
1.80
1.60
1.87
1.64
1.45
1.34
3.18
2.59
2.17
1.81
1.61
3.60
3.08
2.56
2.19
1.94
3.78
3.29
2.73
2.29
2.02
3.81
3.32
2.76
2.32
2.05
3.83
3.34
2.78
2.34
1.93
1.63
1.38
1.24
3.12
2.58
2.33
1.86
1.60
3.52
3.11
2.66
2.07
1.77
3.76
3.36
2.85
2.15
1.83
3.79
3.39
2.88
2.18
1.86
3.81
3.41
2.90
2.20
1,000
0.50
1.50
2.07
1.88
a lateral pressure must also be accounted for, and is given by Equation 2.21. The
stream flow velocity, V, must be in feet per second, and the resulting pressure, p, is
in kips per square foot.
p=
C DV 2
1, 000
(2.20)
p=
CLV 2
1, 000
(2.21)
45
Loads on Bridges
TABLE 2.18
Mississippi Embayment Site Factor Fv from Moon et al. (2016) – Site Class E
S1
NEHRP
Upland
Lowland
30
H, m
0.10
3.50
3.00
3.06
30
30
30
30
100
100
100
100
100
200
200
200
200
200
300
300
300
300
300
500
500
500
500
500
1,000
1,000
1,000
1,000
0.20
0.30
0.40
0.50
0.10
0.20
0.30
0.40
0.50
0.10
0.20
0.30
0.40
0.50
0.10
0.20
0.30
0.40
0.50
0.10
0.20
0.30
0.40
0.50
0.10
0.20
0.30
0.40
3.20
2.80
2.40
2.40
3.50
3.20
2.80
2.40
2.40
3.50
3.20
2.80
2.40
2.40
3.50
3.20
2.80
2.40
2.40
3.50
3.20
2.80
2.40
2.40
3.50
3.20
2.80
2.40
2.58
2.15
1.85
1.64
3.90
3.38
2.58
2.27
1.95
4.90
4.06
3.14
2.82
2.32
5.26
4.11
3.14
2.82
2.42
5.26
4.21
3.14
2.84
2.45
5.26
4.21
3.14
2.84
2.36
1.78
1.33
1.08
4.40
3.48
2.77
2.26
1.82
4.80
3.74
3.17
2.63
2.13
5.06
3.93
3.35
2.68
2.33
5.50
4.40
3.48
2.83
2.54
5.50
4.52
3.65
2.98
1,000
0.50
2.40
2.47
2.65
The longitudinal drag coefficient, CD, depends on the pier type and shape.
•
•
•
•
Semi-circular nose pier, CD = 0.7
Square-ended pier, CD = 1.4
Debris lodged against pier, CD = 1.4
Wedge-nosed pier with nose angle 90 degrees or less, CD = 0.8
The lateral drag coefficient, CL, is a function of the angle, θ, between the flow direction and the longitudinal pier axis.
46
LRFD Bridge Design
θ = 0 degrees, CL = 0.0
θ = 5 degrees, CL = 0.5
θ = 10 degrees, CL = 0.7
θ = 20 degrees, CL = 0.9
θ ≥ 30 degrees, CL = 1.0
For additional guidance on stream flow pressure and other water-related loads, refer
to Section 3.7 of the LRFD-BDS.
2.10 SOLVED PROBLEMS
Problem 2.1
Estimate the braking forces on each pier for a three-span bridge carrying
two lanes of opposing traffic. However, traffic projections indicate that the
bridge may be converted to a one-directional highway in the future with
the construction of an adjacent, dual bridge. Span lengths are 168 ft, 240 ft,
and 168 ft, for a total bridge length of 576 feet. Although the span arrangement is symmetrical, the pier heights are different: 28 feet at Pier 1 and 41
feet at Pier 2. For this example, assume that expansion joints exist at both
abutments and the entire braking force is carried by the piers. Consider two
trucks per loaded lane with the 10% reduction when such a condition is used
for intermediate support reactions.
Assume that the bridge will be one-directional in the future and design
for two lanes of braking forces with the corresponding multi-presence factor, m = 1.0. In addition, check the one-loaded-lane with multi-presence
m = 1.2.
Problem 2.2
Consider a curved bridge with the radius of curvature of the traffic lane in
question equal to R = 730 feet, superelevation, SE = 0.08 ft/ft, and design
speed of 50 mph. The standard design truck with h = b = 6 feet will be used
for the calculations. Determine the distribution of live load to the wheel
lines and the centrifugal force, FCE.
Problem 2.3
A 45 ft 3 inch wide, five-span bridge crosses a lake. Span lengths are 270
ft, 335 ft, 335 ft, 335 ft, and 270 ft for a total length of 1,545 ft. The bridge
is located in an area with a mapped 3-second gust wind speed equal to 115
mph. The superstructure is 12.75 ft deep, measured from the top of the
parapet to the bottom of the girder. Pier number 2 elevations are listed in
the bullet points below. The pier consists of three 11 ft diameter columns.
For the case of wind loading perpendicular to traffic, determine the design
superstructure and substructure wind loads for Strength III, Strength V,
Service I, and Service IV limit states.
• lake bed elevation = 520 ft
• normal pool elevation = 648 ft
• finished grade elevation = 733 ft
47
Loads on Bridges
Problem 2.4
Suppose the Project Bridge is to be constructed in Martin, Tennessee on
State Route TN-43 over State Route TN-22. Soil borings (hypothetical, not
actual for Martin) indicate a subsurface profile as shown below. Determine
the control points of the AASHTO design response spectrum, AS, SDS, SD1,
TS, To, and TL.
• Layer 1, Cohesionless, 65 ft thick, Ni = 12 blows/ft
• Layer 2, Cohesionless, 12 ft thick, Ni = 18 blows/ft
• Layer 3, Cohesive, 12 ft thick, Ni = 27 blows/ft
• Layer 4, Cohesionless, 16 ft thick, Ni = 39 blows/ft
Problem 2.5
A 1,545 ft long, steel I-girder bridge consists of five spans: 270 ft, 335 ft,
335 ft, 335 ft, and 270 ft for a total length of 1,545 ft. Both abutments have
expansion joints. Successive pier heights are 47 ft, 47 ft, 93 ft, and 73 ft,
beginning at Pier No. 1. Identical column cross sections are used at each
pier and the bridge is located in a cold climate. Determine (a) the expansion joint requirements and (b) the movement induced on each pier due to
uniform temperature change (TU) effects.
Problem 2.6
Determine the shear and moment at tenth-points of an interior girder for
the steel girder option of the Project Bridge for the following loads. Use a
flange width equal to 16 inches for concrete in the haunch. Take the girder
spacing, S = 9 ft 3 inches and use a W40 × 215 girder. Include 7.5% of the
girder self-weight in DC1 to account for miscellaneous items. Each parapet
weighs 0.400 klf.
• Dead load on non-composite section (DC1)
• Dead load on composite section (DC2)
• Dead load wearing surface (DW)
PROBLEM 2.1
TEH
1/1
Case 1. 0.90 × (25% of two design trucks in each lane) with m = 1.0.
FBR = 0.90 éë0.25 ( 72 k ´ 2 trucks ) ùû ( 2 lanes ) = 64.8 kips
Case 2. 0.90 × (5% of two design trucks in each lane plus uniform lane load) with
m = 1.0.
FBR = 0.90 éë0.05 ( 0.640 klf ´ 576 ft + 72 k ´ 2 trucks ) ùû ( 2 lanes ) = 46.1 kips
Case 3. 0.90 × (5% of two design trucks in one lane plus uniform lane load) with
m = 1.2.
FBR = 0.90 éë0.05 ( 0.640 klf ´ 576 ft + 72 k ´ 2 trucks ) ùû (1 lane ) (1.2 ) = 27.7 kips
48
LRFD Bridge Design
Case 4. 0.90 × (25% of two design trucks in one lane) with m = 1.2.
FBR = 0.90 éë0.25 ( 72 k ´ 2 trucks ) ùû (1 lane )(1.2 ) = 38.9 kips
Case 1 controls and the total braking force of 64.8 kips can be distributed to the
piers. Set the relative stiffness, K R = 1.0 for Pier 2, the tallest pier. Recognize that
elastic stiffness is proportional to the inverse of the height cubed.
3
æ 41 ö
K R1 = 1.000 ç ÷ = 3.140
è 28 ø
3.140
æ
ö
FBR - P1 = 64.8 ç
÷ = 49.1 kips
3
.
140
+
1
.
000
è
ø
1.000
æ
ö
FBR - P 2 = 64.8 ç
÷ = 15.7 kips
è 3.140 + 1.000 ø
PROBLEM 2.2
TEH
1/1
After converting the design speed to v = 73.33 feet per second, find C = 0.305 (with
f = 4/3).
For SE = 0.08, ϕ = 0.07983 radians. Solving the equations in the text gives the
following:
FN 2 W = 0.735 ( the outside wheel normal force is 73.5% of the truck weight )
FN1 W = 0.286 ( the inside wheel normal force is 28.6% of the trruck weight )
FT1 W = 0.063 ( the inside wheel traction force is 6.3% of the truck weight )
FT 2 W = 0.161 ( the outside wheel traction force is 16.1% of thee truck weight )
FV 2 W = 0.720 ( the outside wheel vertical force is 72.0% of thee truck weight )
FV1 W = 0.280 ( the inside wheel vertical force is 28.0% of the truck weighht )
FCE = 0.305 ´ 72 = 22.0 kips per lane
49
Loads on Bridges
PROBLEM 2.3
TEH
1/4
50
PROBLEM 2.3
LRFD Bridge Design
TEH
2/4
51
Loads on Bridges
PROBLEM 2.3
TEH
3/4
52
LRFD Bridge Design
PROBLEM 2.3
TEH
4/4
PROBLEM 2.4
TEH
1/4
Based on all four layers, determine the average blow count.
( N )1 =
å di
100
=
= 14.8 < 15 ® Site Class E
65 12 7 16
di
å
+ +
+
N i 12 18 27 39
Using only the cohesionless layers, determine the average blow count.
( N )2 =
å di
93
=
= 14.3 < 15 ® Site Class E
65 12 16
di
å
+ +
N i 12 18 39
Use Site Class E site factors, FPGA, Fa, and Fv.
Use the Unified Hazard Tool (UHT) at the United States Geological Survey
(USGS) web site (https://earthquake​.usgs​.gov​/hazards​/interactive/) and determine the
7% in 75-year probability of exceedance bedrock PSA values for Martin, Tennessee.
MRI =
-t
-75
=
= 1, 033years
ln (1 - PE ) ln (1 - 0.07 )
The UHT online application may also be used to pinpoint the project coordinates.
Obtain PSA at 0 seconds (PGA), at 0.2 seconds (SS), and at 1.0 second (S1) at
the B/C boundary.
53
Loads on Bridges
Though not required for bridges to be designed using a dynamic response
spectrum analysis, hazard de-aggregation may also be obtained at the site to
reveal magnitude and distance pairs characterizing the hazard at the site. Such
de-aggregation will prove essential when designing using response history analysis on computer models subject to ground motion accelerograms.
PROBLEM 2.4
TEH
2/4
54
LRFD Bridge Design
PROBLEM 2.4
TEH
3/4
Using the 2014, version 4.2.0 dataset at the USGS UHT gives the following
values for the control point bedrock PSA values:
PGA = 0.4472 g at T = 0.00 seconds
SS = 0.7607 at T = 0.20 seconds
S1 = 0.2070 g at T = 1.00 seconds
From Tables in the text, obtain code-based site factors using interpolation where
required.
FPGA = 0.900 for Site Class E with PGA > 0.400
Fa = 1.20 - (1.20 - 0.90 )
0.7607 - 0.75
= 1.187
1.00 - 0.75
Fv = 3.20 - ( 3.20 - 2.80 )
0.2070 - 0.20
= 3.172
0.30 - 0.20
PROBLEM 2.4
TEH
4/4
The surface, design response spectrum control points may now be determined.
As = 0.4472 ´ 0.900 = 0.4025
SDS = 0.7607 ´1.187 = 0.9029
SD1 = 0.2070 ´3.172 = 0.6566
0.6566
= 0.727 seconds
0.9029
To = 0.20 ´ 0.727 = 0.145 seconds
TS =
55
Loads on Bridges
For this problem, de-aggregation reveals a modal M, R = 7.76, 50 km for all three
control points. The modal M, R combination is the one most likely to produce
ground motion exceeding the design value.
The ATC Hazard by Location tool may be used to find that the transition
period, TL, = 12 seconds for the project site.
PROBLEM 2.5
TEH
1/3
56
PROBLEM 2.5
LRFD Bridge Design
TEH
2/3
57
Loads on Bridges
PROBLEM 2.5
TEH
3/3
PROBLEM 2.6
TEH
1/4
Deck weight = (8.25/12) (9.25) (0.150) = 0.954 klf per interior girder
Haunch weight = (2/12) (16/12) (0.150) = 0.033 klf per interior girder
Girder weight = 0.215 × 1.075 = 0.226 klf per interior girder
• DC1 = 0.954 + 0.033 + 0.226 = 1.213 klf per interior girder
Parapet weight = 2 × 0.400/4 girders = 0.200 klf per interior girder
• DC2 = 0.200 klf per interior girder
58
LRFD Bridge Design
Overlay = 32 ft × 0.035 ksf/4 girders = 0.280 klf per interior girder
• DW = 0.280 klf per interior girder
Use VisualAnalysis to create a two-span, continuous beam model:
CEE4380 ​-Problem​- 02​- 06​​.vap
Notes:
• It is common practice to assume that DC1 weight applied to a girder includes
the self-weight of the girder plus the tributary width of the deck slab.
• Overlay load is commonly distributed equally to all girders.
• Some engineers distribute parapet loads equally to all girders, but it is not
uncommon to find offices in which parapet loads are assumed to be carried only by the exterior and adjacent interior girders.
PROBLEM 2.6
TEH
Due to symmetry, it is only necessary to print results for one span.
2/4
59
Loads on Bridges
PROBLEM 2.6
DC1
DC2
TEH
3/4
60
LRFD Bridge Design
PROBLEM 2.6
TEH
4/4
DW
2.11 EXERCISES
E2.1.
Determine the braking forces required for design of the pier for the Project
Bridge. Assume that the abutments are integral and of equal stiffness to
the pier. Assume further that the bridge will become one-directional in the
future with a dual structure added.
E2.2.
Model the two spans for the concrete girder option of the Project Bridge
and determine shears and moments due to DC1 (dead load of components,
non-composite), DC2 (dead load of components, composite) and DW (dead
load of utilities and overlay) effects on an interior girder. Plot the shear and
moment diagrams. Assume each BT-54 girder supports a tributary width of
deck for DC1 calculations. Assume parapet (0.400 klf per parapet) loading
and wearing surface (35 psf) loading are distributed equally to all four girders. Take the girder spacing, ‘S’ equal to 9 ft 3 inches.
E2.3.
Suppose the Project Bridge is to be constructed in Union City, Tennessee
on State Route TN-22 over N. Clover Street. Soil borings (hypothetical, not
actual for Union City) indicate a subsurface profile as shown in Table E2.3
below. Determine the control points of the AASHTO design response spectrum, AS, SDS, SD1, TS, To, and TL. Consider two cases: (a) using AASHTO
site factors and (2) using the Hashash et al. (2008) uplands site factors. Plot
the surface spectra for each case and compare. Use an estimated subsurface
profile depth of 500 meters.
61
Loads on Bridges
TABLE E2.3
Soil Profile for Exercise E2.3
Layer
Thickness, di (feet)
Blow Count, Ni (blows/ft)
1, cohesionless
5
7
2, cohesionless
3, cohesive
4, cohesionless
5, cohesionless
6, cohesionless
7, cohesionless
8, cohesionless
9, cohesive
5
13
10
10
10
10
10
17
8
36
17
24
44
51
57
69
10, cohesionless
10
32
E2.4.
A continuous steel girder bridge consists of 234-ft end spans with five 300ft interior spans for a total bridge length of 1,968 feet. Pier heights are 47
feet, 47 feet, 73 feet, 73 feet, 31 feet, and 17 feet. Expansion joints and
bearings are used at both abutments only. Pier cross-section geometry is
identical at all piers. The bridge is in a cold climate. Estimate the expansion
joint requirements at Abutment 1 and Abutment 2.
E2.5.
Figure E2.5 is a cross section, looking forward, of a curved bridge with centerline radius equal to 800 feet. The bridge curves to the right. Determine
the centrifugal force on the pier for the two-span bridge. Spans are equal
and the total bridge length is 280 feet. The design speed is 40 mph, and the
bridge is one-directional.
FIGURE E2.5
62
LRFD Bridge Design
E2.6.
A bridge in Jackson, Tennessee is considered critical and will be designed
for the 2,500-year ground shaking rather than the standard 1,033-year
ground shaking. SS = 0.7905 and S1 = 0.2457 for the 2500-year MRI. The
site is an uplands Class D site. Assess the various options for site factors and
develop a proposed design response spectrum for the project.
3
Load Combinations
and Limit States
Chapter 2 presents a discussion of the basic load and resistance factor design (LRFD)
relationship. This discussion includes the idea of both load (γ) factors and resistance
(ϕ) factors, which will be further elaborated on in the current chapter.
The American Association of State Highway and Transportation Officials
(AASHTO) LRFD Bridge Design Specifications (BDS) design philosophy incorporates various limit states into design criteria. The limit states include:
• Strength I limit state: normal traffic with no wind load
• Strength II limit state: a means for owners to specify special design vehicles
• Strength III limit state: maximum design wind load on the bridge with no
traffic
• Strength IV limit state: appropriate for bridges dominated by dead load
effects
• Strength V limit state: an intermediate condition between Strength I and
Strength III
• Service I limit state: normal operation with a wind velocity equal to 70 mph
• Service II limit state: control of yielding in steel and slip in slip-critical
connections
• Service III limit state: tension in prestressed concrete superstructures
• Service IV limit state: crack control in prestressed concrete columns
• Fatigue I limit state: infinite life, load-induced fatigue
• Fatigue II limit state: finite life, load-induced fatigue
• Extreme Event I limit state: earthquake loading with collapse prevention
• Extreme Event II limit state: blast, ice, vehicle collision, and vessel collision
For a typical I-girder bridge, it would be necessary to evaluate Strength I, III, and
V limit states. Strength IV evaluation is necessary for bridges with unusually high
dead load-to-live load ratio. Service limit states applicable for precast, prestressed
concrete I-girder bridges include Service I and Service III. For steel I-girder bridges,
Service limit states I and II apply.
Load sources at the various limit states in AASHTO are characterized as either
permanent or transient, and are defined as follows:
Permanent Loads
• CR = force effects due to creep
• DD = downdrag force
DOI: 10.1201/9781003265467-3
63
64
LRFD Bridge Design
•
•
•
•
•
•
•
•
DC = dead load of structural components and nonstructural attachments
DW = dead load of wearing surfaces and utilities
EH = horizontal earth pressure load
EL = miscellaneous locked-in force effects from the construction process
ES = earth surcharge load
EV = vertical pressure from dead load of earth fill
PS = total prestress forces for Service limit states
SH = force effects due to shrinkage
Transient Loads
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
BL = blast loading
BR = vehicular braking force
CE = vehicular centrifugal force
CT = vehicular collision force
CV = vessel collision force
EQ = earthquake load
FR = friction load
IC = ice load
IM = vehicular dynamic load allowance
LL = vehicular live load
LS = live load surcharge
PL = pedestrian live load
SE = force effect due to settlement
TG = force effect due to temperature gradient
TU = force effect due to uniform temperature
WA = water load and stream pressure
WL = wind on live load
WS = wind load on structure
Equation 3.1 is the basic LRFD-based design equation, with ductility (ηD), redundancy (ηR), and importance (ηI ) modifiers all equal to 1.0. This will be the case for
most routine bridge designs. Refer to Chapter 2 for additional details on the load
modifiers, and situations when values equal to 1.0 may not be appropriate.
Qn = å g iQi £ f Rn
(3.1)
Tables 3.1 through ​​3.4 summarize load factors (γ) for the specified limit states. Note
that, for many limit states, multiple factors are listed for TU-loading (uniform temperature variation). The smaller value is intended to be used for force-related actions,
and the larger value for deformation-related actions.
Table 3.5 lists the variable load factors, γp, for the Strength limit state load combinations. For variable load factors related to thermal gradient (TG) and support
settlement (SE), the reader is referred to Chapter 3 of the LRFD-BDS.
For the Extreme Event II limit state, only one extreme load at a time (BL, IC, CT,
or CV) is considered.
65
Load Combinations and Limit States
TABLE 3.1
Strength Limit State Load Factors
DC
DD
DW
EH
EV
ES
EL
PS
CR
SH
LL
IM
CE
BR
PL
LS
WA
WS
WL
FR
TU
TG
SE
Strength I
γp
1.75
1.00
–
–
1.00
0.50/1.20
γTG
γSE
Strength II
Strength III
Strength IV
γp
γp
γp
1.35
–
–
1.00
1.00
1.00
–
1.00
–
–
–
–
1.00
1.00
1.00
0.50/1.20
0.50/1.20
0.50/1.20
γTG
γTG
–
γSE
γSE
–
Strength V
γp
1.35
1.00
1.00
1.00
1.00
0.50/1.20
γTG
γSE
Limit State
TABLE 3.2
Service Limit State Load Combinations
Limit State
DC
DD
DW
EH
EV
ES
EL
PS
CR
SH
LL
IM
CE
BR
PL
LS
WA
WS
WL
FR
TU
TG
SE
Service I
1.00
1.00
1.00
1.00
1.00
1.00
1.00/1.20
γTG
γSE
Service II
Service III
1.00
1.00
1.30
γLL
1.00
1.00
–
–
–
–
1.00
1.00
1.00/1.20
1.00/1.20
–
γTG
–
γSE
Service IV
1.00
–
1.00
1.00
–
1.00
1.00/1.20
–
1.00
A subset of LRFD-BDS resistance factors for steel elements and concrete elements is summarized in Tables 3.6 and 3.7.
Exceptions to Tables 3.6 and 3.7 are made at the Extreme Event limit states. At
the Extreme Event limit state, resistance factors are 1.00 for failure modes other than
those shown for ASTM F 3125 bolts and shear connectors.
For concrete elements, it is necessary to determine whether tension-control or
compression-control governs the resistance factor.
66
LRFD Bridge Design
TABLE 3.3
Extreme Event Limit State Load Combinations
Limit State
DC
DD
DW
EH
EV
ES
EL
PS
CR
SH
LL
IM
CE
BR
PL
LS
WA
FR
EQ
BL
IC
CT
CV
Extreme Event I
1.00
γEQ
1.00
1.00
1.00
–
–
–
–
Extreme Event II
1.00
0.50
1.00
1.00
–
1.00
1.00
1.00
1.00
TABLE 3.4
Fatigue Limit State Load Combinations
Limit State
LL
IM
CE
Fatigue I
1.75
Fatigue II
0.80
TABLE 3.5
Variable Load Factors
Maximum γp
Minimum γp
DC – Strength I, II, III, V
1.25
0.90
DC – Strength IV
DW
EH – Active
EH – At Rest
EV – Overall Stability
EV – Retaining Walls & Abutments
ES
PPC with Refined Losses/Elastic Gains
1.50
1.50
1.50
1.35
1.00
1.35
1.50
0.90
0.65
0.90
0.90
NA
1.00
0.75
Load and Limit State
All other PPC
γLL = 1.00
γLL = 0.80
67
Load Combinations and Limit States
TABLE 3.6
Resistance Factors for Steel Elements
Flexure
ϕf = 1.00
Shear
Axial compression, steel only
Flexure with axial compression in CFSTs
Axial compression in composite columns
Net section tensile fracture
Gross section tensile yielding
Bearing on milled surfaces
Bolts bearing on material
Shear connectors
ASTM F 3125 bolts in tension
ASTM F 3125 bolts in shear
ASTM F1554 anchor rod in tension
ASTM F1554 anchor rod in shear
Block shear
Shear rupture in connected elements
Web crippling
Weld metal, complete penetration welds
Weld metal, fillet welds
ϕv = 1.00
ϕc = 0.95
ϕc = 0.90
ϕc = 0.90
ϕu = 0.80
ϕy = 0.95
ϕb = 1.00
ϕbb = 0.80
ϕsc = 0.85
ϕt = 0.80
ϕs = 0.80
ϕt = 0.80
ϕs = 0.75
ϕbs = 0.80
ϕvu = 0.80
ϕw = 0.80
ϕe1 = 0.85
ϕe2 = 0.80
Resistance during pile driving
ϕ = 1.00
TABLE 3.7
Resistance Factors for Normal Weight Concrete Elements
Tension-controlled reinforced concrete sections
ϕ = 0.90
Tension-controlled prestressed concrete with bonded strand
Shear and torsion in reinforced concrete
Compression-controlled sections with spirals or ties
Bearing on concrete
ϕ = 1.00
ϕ = 0.90
ϕ = 0.75
ϕ = 0.70
Resistance during pile driving
ϕ = 1.00
Tension-controlled reinforced concrete sections are defined as sections with a net
tensile strain in the extreme layer of tensile reinforcement, εt, greater than or equal
to the tension-controlled strain limit, εtl, when the concrete strain reaches a value of
0.003.
Compression-controlled reinforced concrete sections are defined as sections with
a net tensile strain in the extreme layer of tensile reinforcement, εt, less than or equal
to the compression-controlled strain limit, εcl, when the concrete strain reaches a
value of 0.003.
68
LRFD Bridge Design
The tension-controlled strain limit, εtl, is determined as follows:
• εtl = 0.005 for reinforcement with f y ≤ 75 ksi
• εtl = 0.008 for reinforcement with f y = 100 ksi
• εtl is determined by linear interpolation for reinforcement with
75 < f y < 100 ksi
The compression-controlled strain limit, εcl, is determined as follows.
• εcl = f y/Es, but not > 0.002, for reinforcement with f y ≤ 60 ksi
• εcl = 0.004 for reinforcement with f y = 100 ksi
• εcl is determined by linear interpolation for reinforcement with
60 < f y < 100 ksi
For sections with net tensile strain in the extreme layer of tension reinforcement
between εcl and εtl, linear interpolation, as given below in Equation 3.2, is used to
determine the appropriate resistance factor:
0.75 £ f = 0.75 +
0.15 ( e t - e cl )
£ 0.90
e tl - e cl
(3.2)
For concrete stress-strain profiles, the AASHTO LRDS-BDS permits the use of
“rectangular, parabolic, or any other shape that results in a prediction of strength
in substantial agreement with test results” (Section 5.6.2). For hand calculations, the traditional rectangular assumption is often used. For section analysis
by computer, with Response 2000, for example, a parabolic assumption is not
uncommon.
3.1 SOLVED PROBLEMS
Problem 3.1
Figure P3.1 represents a single girder line of a three-span continuous bridge.
The reactions for a single girder at abutment number 1 and pier number 1
are summarized in Table P3.1. Determine the Strength limit state girder
reactions at abutment number 1 and pier number 1. Negative reactions indicate uplift.
FIGURE P3.1 Problem 3.1.
69
Load Combinations and Limit States
TABLE P3.1
Reactions for Problem P3.1
Abutment 1 Reaction
Pier 1 Reaction
DC
Load Case
15.1 kips
174.6 kips
DW
LL+IM (Maximum)
2.6 kips
72.3 kips
29.6 kips
143.8 kips
LL+IM (Minimum)
−26.6 kips
−5.7 kips
Problem 3.2
Section 6.6.1.2. of the AASHTO LRFD-BDS requires that fatigue be
investigated whenever unfactored compressive dead load stress is less
than Fatigue I limit state tensile stress for the detail under consideration.
At a particular point in a plate girder, long-term composite properties are
used for composite dead load stress calculations (DC2 and DW), girder
properties alone are used for non-composite dead load effects (DC1), and
short-term composite properties are used for live load and fatigue stress calculations. Determine whether flexural fatigue stress range calculations need
to be made for the top flange and for the bottom flange. Properties (section
moduli, Table P3.2a) and unfactored moments (Table P3.2b) are provided.
Positive moment causes compression in the top flange, tension in the bottom flange.
TABLE P3.2A
Section Properties for Problem P3.2
Section
Stop, in3
Sbott, in3
Girder alone
1,088
1,088
Short-term composite
14,261
1,470
Long-term composite
3,867
1,359
TABLE P3.2B
Moments for Problem P3.2
Load Case
Moment, ft-kips
DC1
181
DC2
DW
Fatigue + IM (Maximum)
29
40
296
Fatigue + IM (Minimum)
−175
70
LRFD Bridge Design
Problem 3.3
The top of a circular, cantilever pier column supporting a curved bridge is
subjected to the actions summarized in Table P3.3. The single-post column
height, H, is 82 feet from the point of load application to fixity at the base.
Determine Strength I, Strength III, Strength V, and Service I axial load,
TABLE P3.3
Column Loads for Problem 3.3
Load Case
P, kips
VT, kips
VL, kips
MT, ft-k
ML, ft-k
DC
1,922
–
–
–
89
272
744
–
–
–
–
–
–
33
–
89
18
–
–
–
36
22
8
–
–
–
432
–
–
–
3,929
396
–
801
324
–
23
30
–
–
DW
LL+IM
CE
BR
WS
WL
TU
FIGURE P3.3
Problem 3.3.
71
Load Combinations and Limit States
moment, and shear acting on the base of the column. Values for WS in the
table are for the Strength III limit state wind speed of 115 mph. Values for
other limit states should be adjusted accordingly (Figure P3.3):
• P = axial compression
• V T = shear in the transverse direction, perpendicular to traffic
• VL = shear in the longitudinal direction, parallel to traffic
• MT = moment about a transverse axis
• ML = moment about a longitudinal axis
Problem 3.4
A 48-inch square reinforced concrete bridge column consists of 20 #10
bars distributed uniformly around the perimeter. Clear cover is 2.5 inches.
Concrete and steel strengths are f’c = 4,000 psi and fy = 60 ksi. The momentto-shear ratio for the column is 20 feet for the limit state under investigation.
Determine the appropriate resistance factor, ϕ, and the design resistance,
ϕMn, for the following cases:
a) Axial load, Pu = 0
b) Axial load, Pu = 2,230 kips compression
PROBLEM 3.1
TEH
1/1
72
PROBLEM 3.2
LRFD Bridge Design
TEH
1/1
73
Load Combinations and Limit States
PROBLEM 3.3
TEH
1/4
74
PROBLEM 3.3
LRFD Bridge Design
TEH
2/4
75
Load Combinations and Limit States
PROBLEM 3.3
TEH
3/4
76
PROBLEM 3.3
LRFD Bridge Design
TEH
4/4
77
Load Combinations and Limit States
PROBLEM 3.4
TEH
1/4
78
PROBLEM 3.4
LRFD Bridge Design
TEH
2/4
79
Load Combinations and Limit States
PROBLEM 3.4
Pu = 0 kips:
TEH
3/4
80
LRFD Bridge Design
PROBLEM 3.4
TEH
4/4
Pu = 2,230 kips compression (“-“in Response 2000)
3.2 EXERCISES
Exercise 3.1.
Figure E3.1 shows a section of bridge deck overhang with loads from various sources. Determine the Service I, Strength I, and Extreme Event II limit
state tension (T) and moment (M) in the bridge deck. The parapet is 1 ft
wide over the entire parapet height.
• PDC = 0.400 klf from the parapet
• PLL = 3.5 klf from live load (impact not included)
Load Combinations and Limit States
FIGURE E3.1
Exercise E3.1.
• PCT = 6.2 klf from vehicular collision
• wDW = 35 psf from future overlay allowance
• wDC = deck weight of 9-inch-thick concrete (150 pcf)
Exercise 3.2.
Figure E3.2 depicts the loading on an intermediate pier for a continuous
span bridge. The loads are given as follows:
Vertical reactions:
• R DC = 407 kips per girder (same for interior and exterior girders)
• R DW = 79 kips per girder (same for interior and exterior girders)
• R LL = 160 kips per lane from uniform live load
• R LL = 115 kips per lane from truck live load
Wind loads:
• V1WS = 63 kips for Strength III limit state
• V1WS = 30 kips for Strength V limit state
• V1WS = 23 kips for Service I limit state
• V1WS = 35 kips for Service IV limit state
• V2WS = 6 kips for Strength III and Strength V limit states
• PWS = 144 kips for Strength III limit state
• PWS = 72 kips for Service IV limit state
81
82
LRFD Bridge Design
FIGURE E3.2 Exercise E3.2.
Seismic load: VEQ = 496 kips
Determine the required shear, axial force, and moment at the base of the
pier column for each of the following limit states:
• Strength I
• Strength III
• Strength V
• Service I
• Service IV
• Extreme Event I (only one lane of live load and γEQ = 1.0)
4
Deck and Parapet Design
Traditional deck design is accomplished assuming the deck to be a continuous beam
supported at the girders, with primary reinforcement placed transverse to the direction of traffic. This requires establishing an effective strip width of deck for resisting
live load. The equivalent strip method of deck design is covered in detail in Section
4.6.2 of the 9th edition of the AASHTO LRFD BDS. For overhang deck design, the
wheel load from the design truck is to be placed 1 foot from the face of the parapet.
For all other considerations (for example, live load distribution), the wheel load is
placed 2 feet from the parapet face.
Section 9.7.3.2 of the AASHTO LRFD BDS requires distribution reinforcement in
the bottom of girder-bridge decks in the longitudinal direction, with main deck reinforcement transverse to the direction of traffic. Such distribution reinforcement is to
be determined as a percentage, p, of the primary (bottom transverse) reinforcement,
as given by Equation 4.1, a function of the effective span, S. For prestressed and steel
girder bridges, S may conservatively be taken to be equal to the girder spacing.
p=
220
£ 67
S
(4.1)
Equivalent strip widths are summarized in Table 4.1 for cast-in-place concrete decks
on steel or concrete girders. The girder spacing, S, is used to determine interior strip
widths. The distance, X, from the load to the point of support is used to determine
strip width in the deck overhang. Note that both X and S must be expressed in feet,
while the resulting strip width is in inches. As an alternative design method for deck
overhangs at the Strength limit state, Section 3.6.1.3.4 of the AASHTO LRFD BDS
permits the outer wheel load to be replaced by a uniformly distributed load of 1.0 kip
per foot located 1 foot from the face of the barrier, provided (a) the barrier is continuous and (b) the distance from the exterior girder centerline to the face of the barrier,
de, does not exceed 6 feet.
4.1 PARAPET DESIGN
Parapet design for bridges includes accommodating vehicular crash loads (CT), both
vertical and horizontal. Various test-level parameters used for crash load design of
parapets are summarized in Table 4.2. Horizontal forces, Ft and FL, are applied at
the top of the parapet.
Parapet design includes minimum height requirements, as shown in Table 4.2,
as well as resistance requirements. Parapet resistance, Rw, given in Equation 4.4 for
an interior segment of parapet, or in Equation 4.5 for an end segment, must equal or
exceed Ft.
DOI: 10.1201/9781003265467-4
83
84
LRFD Bridge Design
TABLE 4.1
Equivalent Strip Width
Condition
Equivalent Strip Width, inches
Overhang, top of deck (negative moment)
45 + 10.0X
Interior, bottom of deck (positive moment)
26 + 6.6S
Interior, top of deck (negative moment)
48 + 3.0S
TABLE 4.2
Crash Load (CT) Requirements
Parameter
TL-1
TL-2
TL-3
TL-4
TL-5
TL-6
Ft, transverse (kips)
13.5
27.0
54.0
54.0
124.0
175.0
FL, longitudinal (kips)
Fv, vertical (kips)
Lt and LL (feet)
Lv (feet)
He (minimum, inches)
4.5
4.5
4.0
18.0
18.0
9.0
4.5
4.0
18.0
20.0
18.0
4.5
4.0
18.0
24.0
18.0
18.0
3.5
18.0
32.0
41.0
80.0
8.0
40.0
42.0
58.0
80.0
8.0
40.0
56.0
Minimum rail height, H (ft)
27.0
27.0
27.0
32.0
42.0
90.0
Lc is a critical yield line length, in feet, determined using either Equation 4.2 for
an interior segment of parapet, or Equation 4.3 for an end segment of parapet. H is
the actual height of the parapet, in feet. The moment, Mb, is the flexural resistance
of any beam on top of the parapet, has units of ft-kips, and is often zero. Mw is the
flexural resistance of the parapet about a vertical axis, in units of ft-kips.
The moment, MC, is equal to the flexural resistance of the parapet base about an
axis in the direction of traffic, in units of ft-kips per foot. For parapets with excess
capacity, this provision certainly penalizes the design of the deck overhang reinforcement, as will be evident when design cases for the deck overhang are discussed.
Lt
æ L ö 8 H ( Mb + M w )
+ ç t÷ +
, interior segment
2
Mc
è 2 ø
(4.2)
Lt
æ L ö H ( Mb + M w )
+ ç t÷ +
, end segment
2
Mc
è 2 ø
(4.3)
2
Lc =
2
Lc =
æ
öæ
2
M c L2c ö
Rw = ç
÷ , interior segment
ç 8 Mb + 8 Mw +
÷
H ø
è 2 Lc - Lt ø è
(4.4)
85
Deck and Parapet Design
æ
2
Rw = ç
è 2 Lc - Lt
öæ
M c L2c ö
M
+
M
+
b
w
ç
÷ , end segment
÷
H ø
øè
(4.5)
4.2 DECK OVERHANG DESIGN
Section A13.4 of Appendix A13 in the AASHTO-LRFD-BDS requires that deck
overhangs be designed for three cases.
A. Extreme Event II limit state: horizontal crash forces (CT) with load factor
γCT = 1.00, live load plus impact forces (LLL+IM) with load factor γLL = 0.50,
and dead loads (DC, DW) with a load factor γDC = γDW = 1.00.
B. Extreme Event II limit state: vertical crash forces (CT) with load factor
γCT = 1.00, live load plus impact forces (LLL+IM) with load factor γLL = 0.50,
and dead loads (DC, DW) with a load factor γDC = γDW = 1.00.
C. Strength I limit state: live load plus impact (LL+IM) with load factor
γLL = 1.75, dead loads (DC) with load factor γDC = 1.25, dead load (DW) with
load factor γDW = 1.50.
In bridges with concrete parapets, for design case A, the overhang may be designed
for a tensile force, T, acting simultaneously with the moment, MC. The tension, T, is
given by Equation 4.6 and is expressed in units of kips per foot.
T=
Rw
Lc + 2 H
(4.6)
This approach, although recommended in Section 13 of the AASHTO LRFDBDS, is extremely conservative in that contributions of slab resistance perpendicular to traffic on the overhang are completely ignored, but likely to be significant
in reality. Current research (National Cooperative Highway Research Program –
NCHRP Project 12–119) is aimed at more accurately predicting the behavior of deck
overhangs.
4.3 INTERIOR BAY DECK DESIGN
The design of reinforcement in the deck over interior girders (negative moment,
top mat of steel) and mid-span between girders (positive moment, bottom mat of
steel) is fairly straightforward, compared to overhang design. Appendix A4 of the
AASHTO LRFD BDS provides tables for positive and negative live load moment in
bridge decks, expressed in units of ft-kips per foot. The deck design table already has
multi-presence and impact factors incorporated in the tabulated moments. Simplified
design using the table may be used as long as the following conditions are satisfied:
• the deck is supported by three or more girders;
• the distance between centerlines of exterior girders is no less than 14 feet;
86
LRFD Bridge Design
• the girders are parallel; and
• the overhang must be designed separately as outlined in Section 4.2 above.
If the tables are not available, approximate regression equations may be useful in
the following form shown in Equation 4.7 for cases in which the girder spacing is no
less than 7.5 feet and no greater than 15 feet. Coefficients for the equation are summarized in Table 4.3.
M = a + b ´ S + c ´ S2
(4.7)
Live load moments from the table may be combined, using appropriate load factors,
with dead load moments computed using simplified approximations, such as those
presented in Equations 4.8 and 4.9. S, again, is the girder spacing. The design section
distance, ds, is the distance from the centerline of the girder to the design section for
negative moment and is taken to be equal to one-fourth of the flange width for steel
I-girders, and one-third of the flange width for concrete I-girders.
+
M DC
, DW =
M
DC , DW
wDC , DW S 2
11
(4.8)
wDC , DW ( S - 2ds )
=
9
2
(4.9)
TABLE 4.3
Coefficients for Slab Moment Regression Equation
7.5 feet ≤ S ≤ 9.25 feet
9.25 feet < S ≤ 15 feet
a
b
c
a
b
c
Positive M
3.86548
−0.09095
0.04000
−0.56975
0.89761
−0.01519
Negative M
(ds = 0)
Negative M
(ds = 3 inches)
Negative M
(ds = 6 inches)
Negative M
(ds = 9 inches)
Negative M
(ds = 12 inches)
Negative M
(ds = 18 inches)
−1.35893
1.55905
−0.07238
−10.87797
2.42897
−0.05550
−1.89060
1.49048
−0.06857
−11.08878
2.31935
−0.05105
−1.65375
1.24214
−0.05429
−11.32318
2.21442
−0.04681
0.04821
0.62714
−0.01714
−11.18834
2.05032
−0.04023
−7.45810
2.27333
−0.11429
−8.92069
1.55216
−0.02072
−21.02292
4.80500
−0.23333
−3.14296
0.68492
0.00368
−6.07560
1.19667
−0.02095
−6.27981
1.24680
−0.02410
Negative M
(ds = 24 inches)
87
Deck and Parapet Design
In cases where the cross-sectional geometry lies outside the limits for simplified
design, or for cases where more precise deck moments are needed, a continuous
beam model (supports at the girder centerlines) with overhangs may be developed,
with the moving load consisting of side-by-side vehicles comprised of wheel loads
from the HL-93 truck. The magnitude of such wheel loads would need to be 16 kips
(32-kip axle = 16-kip wheel load) × 1.33 (impact) × 1.2 (multi-presence) = 25.5 kips.
Uniform loads from the HL-93 lane load, the deck self-weight, and the overlay
allowance must be included in such models as well. Resulting moments from the
moving load analysis clearly need to be divided by the effective strip width prior to
combining with dead load effects.
Design flexural resistance, ϕMn, must be greater than the Strength limit staterequired flexural resistance, Mu. Refer to Chapters 2 and 3 for previous discussions
on appropriate load (γ) factors and resistance (ϕ) factors.
To control deck cracking, Section 5.6 of the AASHTO LRFD BDS requires that
bar spacing in the extreme tension layer of reinforcement does not exceed s as given
by Equation 4.10.
s£
700g e
- 2 dc
b s fss
bs = 1 +
dc
0.7 ( h - dc )
(4.10)
(4.11)
βs is the ratio of strain at the extreme tension face to strain in the centroid of the reinforcement layer closest to the tension face, and dc is the distance from the extreme
tension face to the centroid of the reinforcement closest to the tension face. The overall height of the member is h. The Service limit state stress in the reinforcement is fss,
which must not exceed 0.60 × f y. The exposure factor, γe, is 1.00 for Class 1 exposure
conditions and 0.75 for Class 2 exposure conditions. Class 1 exposure corresponds
to an estimated crack width of 0.017 inches. Class 2 exposure corresponds to an
estimated crack width of 0.013 inches, as the exposure factor is directly proportional
to the estimated crack width. Decks are one example of a case where it may be advisable to use Class 2 exposure for crack control, given the susceptibility to corrosion
for typical deck top conditions.
For Strength limit state flexural considerations, bridge decks are typically
tension-controlled elements with a resistance factor, ϕ = 0.90. Nevertheless,
such assumptions should always be verified and any adjustments to resistance
incorporated.
The AASHTO LRFD BDS clear cover requirements are found in Section 5.10.1.
For bridge decks with uncoated reinforcing steel subject to tire stud or chain wear,
the minimum clear cover is 2.50 inches. For such a situation, but with epoxy-coated
reinforcement, the minimum clear cover is 2.00 inches. For the bottom of cast-inplace slabs, the minimum clear cover is 1.00 inch for No. 11 bars and smaller. The
maximum bar size in typical bridge decks is #6 for the outer mats perpendicular
88
LRFD Bridge Design
to traffic. For precast, prestressed girders made continuous and requiring large
amounts of longitudinal steel in the deck to resist negative moments at interior supports, longitudinal bar sizes as large as #9 may be required. Congested reinforcement geometries should be avoided and the minimum spacing between the top and
bottom longitudinal mats of reinforcing must be no less than either (a) 1.00 inch and
(b) the longitudinal bar diameter. For an 8.25-inch- thick bridge deck with #6 transverse bars top and bottom, and with #6 bottom longitudinal bars, the maximum size
of the top longitudinal bar which may be used while maintaining the 1-inch clear
distance, is a #8 bar, assuming 2.50-inches top clear cover and 1.00-inch bottom
clear cover.
Transverse mats of deck reinforcing are typically placed outside the longitudinal
mats to provide advantageous effective depth for flexural design of the transverse
reinforcement. Consideration of compression reinforcement for design of the transverse deck reinforcement, although often ignored, may prove beneficial for analysis
at the Service and Strength limit states.
4.4 SOLVED PROBLEMS
Problem 4.1
For the bridge cross-section shown in Figure P4.1, design the transverse
deck reinforcement and the bottom mat of longitudinal distribution reinforcement. Use f’c = 4 ksi, f y = 60 ksi, and Exposure 2 for the top mat of
transverse reinforcement. The girders are steel, welded-plate girders with
top flange width, bf = 18 inches. Also assess the adequacy of the parapet for
TL-4 criteria. For Extreme Event considerations, parapet resistance values
have been determined to be:
• M W = 20.95 ft-k
• MC = 16.82 ft-k/ft
FIGURE P4.1 Problem P4.1.
Deck and Parapet Design
The parapets weigh 0.38 klf each with a center of gravity located 7.5 inches
from the deck edge. Use 2.5-inches clear cover for top bars and 1-inch clear
cover for bottom bars.
Problem 4.2
Determine M W and MC for the parapet shown in Figure P4.2. Assess the
adequacy of the parapet for various TL criteria. Use Tw = 6 inches, f’c = 5 ksi,
and f y = 60 ksi.
Problem 4.3
A 9-inch-thick bridge deck with 2.50 inches top clear cover and 1.00 inches
bottom clear cover has #5transverse bars spaced at 7 inches on centers, top
and bottom. Concrete strength, f’c equals 4 ksi. Yield stress for the epoxycoated reinforcement is 60 ksi. Determine each of the following for both
positive flexure (tension in the bottom of the deck) and negative flexure
(tension in the top of the deck).
• ϕMn, design flexural resistance at the Strength limit state
• Icr, the cracked section moment of inertia
• fss as a function of Service limit state moment, Mu-SER
• βs
FIGURE P4.2 Problem P4.2.
89
90
PROBLEM 4.1
LRFD Bridge Design
TEH
1/11
91
Deck and Parapet Design
PROBLEM 4.1
TEH
2/11
92
PROBLEM 4.1
LRFD Bridge Design
TEH
3/11
93
Deck and Parapet Design
PROBLEM 4.1
TEH
4/11
94
PROBLEM 4.1
LRFD Bridge Design
TEH
5/11
95
Deck and Parapet Design
PROBLEM 4.1
TEH
6/11
96
PROBLEM 4.1
LRFD Bridge Design
TEH
7/11
97
Deck and Parapet Design
PROBLEM 4.1
TEH
8/11
PROBLEM 4.1
TEH
9/11
98
LRFD Bridge Design
PROBLEM 4.1
TEH
10/11
PROBLEM 4.1
TEH
11/11
99
Deck and Parapet Design
Distribution steel (bottom mat, longitudinal):
p=
220
= 69.6 > 67 ® use 67% of No. 5 @8¢¢
10
Use No. 5 at 12 inches on center
PROBLEM 4.2
TEH
Response 2000 file, “CEE4380​-Problem​- 4​.2​-M​W​.rsp”
1/3
100
PROBLEM 4.2
LRFD Bridge Design
TEH
Response 2000 file, “CEE4380​-Problem​- 4​.2​-M​C​.rsp”
2/3
101
Deck and Parapet Design
PROBLEM 4.2
TEH
3/3
From the Response 2000 analyses:
MW = 37.7 ft-kips
MC = 14.5 ft-k/ft
Consider TL-3 criteria (highest level for which H is at least equal to Hmin):
Ft = 54 kips
Lt = 4.0 feet
H = 27 inches (min), H = 30 inches (actual), OK
For an interior parapet segment:
4
æ 4 ö 8 ( 2.5 ) ( 0 + 37.7 )
+ ç ÷ +
= 9.48 ft
2
14.5
è2ø
2
Lc =
2
öæ
æ
14.5 ( 9.48 )
2
ç 8 ( 37.7 ) +
Rw = ç
÷
ç 2 ( 9.48 ) - 4 ÷ ç
2.5
è
øè
ö
÷ = 110 kipss
÷
ø
For an end segment of the parapet:
4
æ 4 ö ( 2.5 ) ( 0 + 37.7 )
+ ç ÷ +
= 5.24 ft
2
14.5
è2ø
2
Lc =
2
æ
öæ
14.5 ( 5.24 )
2
ç 37.7 +
Rw = ç
÷
ç 2 ( 5.24 ) - 4 ÷ ç
2.5
è
øè
ö
÷ = 61 kips
÷
ø
For both interior and end parapet segments, Rw > Ft → OK.
TL-3 criteria are satisfied for the parapet geometry and reinforcement. TL-4
and beyond require parapet heights greater than that provided and are thus not
satisfied, regardless of any computed Rw value.
102
PROBLEM 4.3
LRFD Bridge Design
TEH
1/4
103
Deck and Parapet Design
PROBLEM 4.3
TEH
2/4
104
PROBLEM 4.3
LRFD Bridge Design
TEH
3/4
Response 2000 file “CEE4380-P04.03” is shown on the following page for the
negative moment analysis.
105
Deck and Parapet Design
PROBLEM 4.3
TEH
4/4
106
LRFD Bridge Design
4.5 EXERCISES
E4.1
Compute interior positive and negative moments in the deck for transverse reinforcement design for the steel girder option of the Project Bridge.
Compute the overhang negative moments. Consider Service I and Strength I
limit states only for this problem (in practice, the Extreme Event limit states
would also have to be considered). Design the top and bottom reinforcement
mats of transverse reinforcing. The following parameters are to be used:
• S = 9 ft 3 inches
• C = 3 ft 4 inches
• f’c = 4 ksi
• f y = 60 ksi
• Clear cover = 2.5 inches (top), 1.0 inch (bottom)
• Exposure 2 for top bars in the deck
• Steel girder flange width = 16 inches
• Use #5 bars
• Parapet weight = 0.400 klf
• Parapet center of gravity (c.g.) is 10 inches from the deck edge
E4.2
Compute interior positive and negative moments in the deck for transverse
reinforcement design for the concrete girder option of the Project Bridge.
Compute the overhang negative moments. Consider Service I and Strength I
limit states only for this problem (in practice, the Extreme Event limit states
would also have to be considered). Design the top and bottom reinforcement
mats of transverse reinforcing. The following parameters are to be used:
• S = 9 ft 3 inches
• C = 3 ft 4 inches
• f’c = 4 ksi
• f y = 60 ksi
• Clear cover = 2.5 inches (top), 1.0 inch (bottom)
• Exposure 2 for top bars in the deck
• Use #5 bars
• Parapet weight = 0.400 klf
• Parapet c.g. is 10 inches from the deck edge
E4.3
Assume a parapet with no excess resistance (Rw = Ft) is used for the Project
Bridge. Compute overhang design moments at the Extreme Event limit
state for the steel girder option of the Project Bridge. Assume that the ratio
Mw/Mc = 2.5, and that Mb = 0. Compare the design moment to the Strength
limit state moments from Exercise 4.1.
5
Distribution of Live Load
Live load distribution factors permit girder design to be completed using a line girder
analysis (i.e., a single, continuous beam model) rather than a 3-dimensional or grillage computer model. This greatly simplifies girder design and the interpretation of
results.
AASHTO includes at least three methods for computing live load distribution
factors:
1. AASHTO equations
2. The lever rule
3. The rigid cross-section method
The decreasing likelihood of simultaneous lanes being loaded in the most disadvantageous arrangement, as the number of loaded lanes increases, is incorporated into
AASHTO live load design through the application of a multi-presence factor, m. The
multi-presence factor is applied as follows (with the exception of the Fatigue limit
state, for which multi-presence factors are not applied):
•
•
•
•
1 loaded lane, m = 1.20
2 loaded lanes, m = 1.00
3 loaded lanes, m = 0.85
4 or more loaded lanes, m = 0.65
Several additional definitions will be necessary prior to presenting the various methods available in the AASHTO BDS for live load distribution in I-girder bridges.
•
•
•
•
•
•
•
•
•
•
•
•
•
S = girder spacing, feet
ts = concrete deck thickness, inches
Kg = longitudinal stiffness parameter, in4
L = span length, feet
e = correction factor
de = distance from centerline of exterior I-girder to face of parapet, feet
EB = Young’s modulus for the girder concrete, ksi
ED = Young’s modulus for the deck concrete, ksi
eg = distance between the centers of gravity of the girder and deck, inches
A = non-composite area of a single I-girder, in2
I = non-composite moment of inertia of a single I-girder, in4
NL = number of loaded lanes
NB = number of beams (girders) in the bridge cross section
DOI: 10.1201/9781003265467-5
107
108
LRFD Bridge Design
Note that the units specified in the definitions must be used in the equations.
Any necessary unit conversion factors have already been incorporated into the
equations.
The value of L to be used in the equations depends on the parameter being calculated, as noted below.
• For positive moments, use L for the span under consideration.
• For negative moment near interior supports between points of contraflexure, use the average of the adjacent spans for L.
• For negative moments other than near interior supports, use L for the span
under consideration.
• For shear, use L for the span under consideration.
• For reactions at exterior supports (typically abutments or piers with expansion joints), use L for the span under consideration.
• For reactions at continuous interior supports (piers), use the average of adjacent span lengths for L.
5.1 AASHTO EQUATIONS
The focus of material presented here is on concrete and steel I-girders. For box girders and other types of superstructures, the reader is referred to Section 4.6.2.2 of the
AASHTO LRFD BDS (AASHTO, 2020).
The distribution factors in the AASHTO equations have the multi-presence factor, m, built in. There is no need to apply m to a distribution factor calculated from
the AASHTO equations. The symbol for the distribution factor in AASHTO is g.
The equations listed in AASHTO, and here, define mg.
Since the multi-presence factor is incorporated into the AASHTO equations, it is
necessary to divide an equation-based, single-lane distribution factor by m = 1.2 for
assessment of Fatigue limit state response (lever rule and rigid cross-section methods
may also be appropriate for Fatigue distribution factors). The multi-presence factor is
not to be applied for the Fatigue limit state.
For prestressed I-girders, the live load distribution factors for moment in an interior girder are given by Equations 5.1 and 5.2. The equations apply for four or more
girders, provided each of the following conditions is satisfied:
3.5 £ S £ 16.0
4.5 £ t s £ 12.0
20 £ L £ 240
10, 000 £ K g £ 7, 000, 000
æ S ö
mg = 0.06 + ç ÷
è 14 ø
0.40
æSö
çL÷
è ø
0.30
æ Kg ö
ç 12 Lt 3 ÷
s ø
è
0.10
, 1-lane loaded
(5.1)
109
Distribution of Live Load
æ S ö
mg = 0.075 + ç
÷
è 9.5 ø
0.60
æSö
çL÷
è ø
0.20
æ Kg ö
ç 12 Lt 3 ÷
s ø
è
(
K g = n I + Aeg2
n=
0.10
, multiiple-lanes
)
E B 1, 820 fcB¢
=
=
E D 1, 820 fcD
¢
(5.2)
(5.3)
fcB¢
¢
fcD
(5.4)
With only three girders, the live load distribution factor for moment in an interior
girder is to be taken as the lesser value of that derived from Equation 5.1, Equation 5.2,
or the lever rule (Section 5.2).
For prestressed I-girders with multiple lanes loaded, the live load distribution factor for moment in an exterior girder is given by Equation 5.6, based on the value of
e from Equation 5.5. For I-girders, the distance, de, in Equation 5.5 is defined as the
horizontal distance, in feet, from the centerline of the exterior I-girder to the face of
the curb or barrier. The equations apply for four or more girders with multiple lanes
loaded and with −1.0 ≤ de ≤ 5.5.
e = 0.77 +
de
9.1
mg = e ( mg )interior
(5.5)
(5.6)
With only one lane loaded, the distribution factor for exterior girder moment is determined by the lever rule. With only three girders, the live load distribution factor for
moment in an exterior girder is to be taken as the lesser of the values obtained from
Equation 5.6 or the lever rule.
For prestressed I-girders, the live load distribution factor for shear in an interior
girder is given by Equations 5.7 and 5.8. The equations apply for four or more girders with:
3.5 £ S £ 16.0
4.5 £ t s £ 12.0
20 £ L £ 240
mg = 0.36 +
S
, 1-lane loaded
25
(5.7)
2.0
mg = 0.2 +
S æ S ö
, multiple-lanes loaded
12 çè 35 ÷ø
(5.8)
110
LRFD Bridge Design
With only three girders, the live load distribution factor for shear in an interior girder
is to be calculated using the lever rule, regardless of the number of lanes loaded.
For prestressed I-girders with multiple lanes loaded, the live load distribution factor for shear in an exterior girder is given by Equation 5.10. The equation is applicable with four or more girders, with multiple lanes loaded, and with -1.0 £ de £ 5.5.
e = 0.60 +
de
10
mg = e ( mg )interior
(5.9)
(5.10)
With only one lane loaded, or with only three girders, the live load distribution factor
for shear in an exterior girder is to be calculated using the lever rule.
The equations presented for prestressed I-girders are also applicable to steel
I-girders with cast-in-place deck. However, with steel girder bridges, the cross-section properties are unknown before design of the girder has been completed, so it
is not generally possible to determine Kg from equations. AASHTO provides for an
approximate distribution factor to be calculated, using Equation 5.11, when girder
properties are not known up front.
æ Kg ö
ç 12 Lt 3 ÷
s ø
è
0.10
ì1.02, steel I-girders
=í
î1.09, concrete I-girderrs
(5.11)
The final design, once the cross-sectional properties are known, should include calculation of (a) the stiffness parameter, Kg, from Equation 5.3, (b) the new distribution factor with modified Kg, and (c) subsequent re-calculation of live load moments,
shears, reactions, and displacements if the modified distribution factor differs significantly from that assumed originally.
Skew effects may be accounted for in live load distribution by the AASHTO
equations. Skew tends to decrease moment and increase shear in exterior girders.
Equation 5.12 gives the amplification factor for girder shear in bridges with skewed
supports, applicable only to shear in the exterior girder at the obtuse corner. The correction factor, CF, is to be applied directly to the calculated distribution factor. For
other skew effects, refer to Section 4.6.2.2 of the AASHTO LRFD BDS. Equation
5.12 is applicable subject to the following limitations:
0° £ q £ 60°
3.5 £ S £ 16.0
20 £ L £ 240
NB ³ 4
111
Distribution of Live Load
0.3
æ 12.0 Lt s3 ö
CF = 1.0 + 0.20 ç
÷ tan q
è Kg ø
(5.12)
5.2 THE LEVER RULE
The lever rule is based on a simplifying assumption that the exterior girder and first
interior girder act as a propped beam. Each wheel line of the design truck carries
one-half of the load (for a straight bridge). The assumption is illustrated in Figure 5.1.
By summing moments abut the first interior girder, the reaction at the exterior girder
corresponds to the lever rule-based distribution factor.
When a distribution factor is calculated using the lever rule, the multi-presence
factor should be applied. An exception is the calculation of distribution factors for
the Fatigue limit state, for which multi-presence factors are never to be applied. The
Fatigue loading is defined as a single truck in a single lane, with no added uniform
live load (as opposed to the live load definition for other limit states).
The distance, x, in Figure 5.1, is taken to be no less than 2 feet for distribution
factor calculations. For deck design, the distance x is taken to be no less than 1 foot.
When the bridge is curved in plan, recall that centrifugal forces result in an
unequal distribution of truck weight between the inside and outside wheel lines (see
Section 2.4, this book). Although often ignored in distribution factor calculations, it
may be advisable to consider this unequal distribution of truck weight in the leverrule analysis for bridges with significant curvature.
FIGURE 5.1 Live load distribution factor by the lever rule.
112
LRFD Bridge Design
5.3 RIGID CROSS-SECTION METHOD
Application of the rigid cross-section method of live load distribution is often necessary for steel girder bridges or any bridge with a cross section which may be relatively rigid, in a torsional manner, through the use of cross-frame-like elements. In
Section 4.6.2.2.2d, the AASHTO LRFD BDS specifies that, for exterior girders in
beam-slab bridges with diaphragms or cross-frames, the live load distribution factor
should also be computed using this method. So, even with concrete girders, if rigid
intermediate diaphragms are present in the bridge, the cross-section method should
be applied. Implicit in the method is the assumption that the cross section rotates
as a rigid body. The live load distribution factor of the rigid cross-section method is
defined by Equation 5.13. NL is the number of loaded lanes, and NB is the number of
beams (girders) in the cross section. The distance from the centerline of the bridge
to the center of a lane is the eccentricity, e, for that lane. This is not to be confused
with the adjustment factor, e, used in determining live load distribution factors for
exterior girders, using the AASHTO equations (see Equations 5.5 and 5.9). The distance from the centerline of the bridge to a girder centerline is the parameter, X, for
that girder. The multi-presence factor, m, needs to be incorporated when performing
a rigid cross-section analysis. Lane locations must be determined to maximize each
girder individually (Figure 5.2).
éN
X åeù
mg = m ê L + ext 2 ú
åX û
ë NB
FIGURE 5.2 Live load distribution factor by the rigid cross-section method.
(5.13)
Distribution of Live Load
113
As is the case with lever-rule calculations, it may be advisable to incorporate
unequal weight distribution between the inner and outer wheel lines due to centrifugal forces in bridges with plan curvature when using the rigid cross-section
method.
5.4 SOLVED PROBLEMS
Problem 5.1
For the steel girder option of the Project Bridge, calculate the live load distribution factors for shear and moment for an exterior girder using all three
available methods. Use the following parameters:
• S = 9 ft 3 inches
• C = 3 ft 4 inches
• W40 × 215 Girders, A = 63.5 in2, I = 16,700 in4
• f’c = 4,000 psi for the deck
Problem 5.2
For the concrete girder option of the Project Bridge, calculate the live load
distribution factors for shear and moment for both interior and exterior girders using all applicable methods. Use the following parameters:
• S = 9 ft 3 inches
• C = 3 ft 4 inches
• BT-54 Girders, A = 659 in2, I = 268,077 in4
• f’c = 4,000 psi for the deck, f’c = 9,000 psi for the girders
Problem 5.3
A steel I-girder bridge is 77-ft wide with 1-ft wide parapets on both edges.
Girder spacing is 10 ft and eight girders are used in the cross section.
Determine the exterior girder live load distribution factor by the rigid
cross-section method for all applicable loaded land cases. In practice, all
other applicable methods (AASHTO equations and lever-rule method)
would have to be determined as well to establish the controlling conditions. The purpose of this problem is to fully illustrate the rigid crosssection method.
Problem 5.4
Using the results from Problem 5.1, establish a design moving load case
for (a) the Fatigue limit state and (b) for all other limit states. Include
any necessary multi-presence and impact factors, but do not include load
factors.
114
PROBLEM 5.1
LRFD Bridge Design
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115
Distribution of Live Load
PROBLEM 5.1
TEH
2/3
116
LRFD Bridge Design
PROBLEM 5.1
TEH
For Fatigue, use the lever rule without multi-presence factors:
( mg )Fat =
0.826 1.2 = 0.688 lanes per girder
3/3
117
Distribution of Live Load
PROBLEM 5.2
TEH
1/3
118
PROBLEM 5.2
LRFD Bridge Design
TEH
2/3
119
Distribution of Live Load
PROBLEM 5.2
TEH
3/3
120
PROBLEM 5.3
LRFD Bridge Design
TEH
1/2
121
Distribution of Live Load
PROBLEM 5.3
TEH
2/2
122
PROBLEM 5.4
LRFD Bridge Design
TEH
1/2
123
Distribution of Live Load
PROBLEM 5.4
TEH
2/2
124
LRFD Bridge Design
5.5 EXERCISES
E5.1
A 284-ft long, two-span bridge consists of four prestressed concrete BT-72
I-girders spaced 8 ft 3 inches apart. The bridge is 31 ft 3 inches wide and
has equal span lengths. The inside face of the parapet is 1 foot from the
outer edge of the bridge deck. Girder concrete has a minimum specified
28-day concrete strength of 10 ksi, while the deck uses 4 ksi concrete. Deck
thickness is 8.25 inches. The haunch (distance from top of girder to bottom
of deck) is 2 inches. The properties may be assumed constant for the entire
bridge length. For the Service and Strength limit states, determine:
• distribution factors for moment and shear for an interior girder
• distribution factors for moment and shear for an exterior girder
E5.2
For the bridge of Exercise E5.1, determine the appropriate distribution factors for shear and moment at the Fatigue limit state for (a) an interior girder
and (b) an exterior girder. Use a continuous beam computer model to generate shear and moment envelopes for the controlling girder for the Fatigue
limit state.
E5.3
A steel I-girder bridge is 57 feet wide. Five girders spaced 12 feet apart
are used for the bridge cross-section. The parapet width is 1 ft 6 inches at
the base. Determine the live load distribution factors for an exterior girder
using the rigid cross-section method. Consider all possible numbers of
loaded lanes.
E5.4
For the steel superstructure option of the Project Bridge, determine the live
load distribution factors for moment and shear in an interior girder. Model
the continuous spans in VisualAnalysis and create moving load combinations to determine shear and moment envelopes for LL+IM corresponding
to the Fatigue limit state (IM = 15%) and all other limit states (IM = 33%).
Plot the envelopes. Verify using the National Steel Bridge Alliance (NSBA)
LRFD Simon software. Use girder spacing S = 9 ft 3 inches and W40 × 215
girders. Deck concrete f’c is equal to 4,000 psi.
E5.5
A 57-ft 6-inch-wide bridge is curved in plan with centerline radius of curvature equal to 750 ft. Parapet width is 1 ft at the base. Girder spacing, S = 12
feet and the overhang dimension, C = 4 ft 9 inches. Design speed is 50 mph.
Incorporate the centrifugal force effect on wheel line distribution and compute live load distribution factors using (a) the lever-rule method and (b) the
rigid cross-section method. Superelevation is 0.08 ft/ft.
6
Steel Welded
Plate I-Girders
Details to be developed and designed for steel girder bridges include each of the
following:
•
•
•
•
•
•
•
•
•
•
•
•
•
girder transitions (flange width and thickness; web depth and thickness)
bolted field splices
cross-frame layout
cross-frame details
bearing stiffeners
shear stiffeners
cross-frame connection stiffeners
shear studs
camber diagrams
web-to-flange weld
girder stability (lateral bracing; torsional bracing)
bearings
pouring sequence
The focus of the material presented in this chapter is on the Strength and Fatigue
limit states. There are many other requirements for steel I-girder bridges, including
those for Constructability and Service limit states. For a detailed coverage of design
information not covered here for steel I-girders, refer to the AASHTO LRFD BDS,
Section 6.10 (AASHTO, 2020).
To qualify as a compact section in positive flexure at the Strength limit state in the
AASHTO LRFD BDS, a steel I-girder must satisfy each of Equations 6.1, 6.2, and
6.3. Parameters in the equations include the depth of the web in compression (Dcp),
the web thickness (t w), the total web depth (D), Young’s modulus (E = 29,000 ksi),
and the yield strength of the compression flange (Fyc).
DOI: 10.1201/9781003265467-6
Fy £ 70 ksi for flanges
(6.1)
2 Dcp
E
£ 3.76
tw
Fyc
(6.2)
D
£ 150
tw
(6.3)
125
126
LRFD Bridge Design
Dcp is the web depth in compression for the plastic stress distribution. Section proportion limits are summarized for I-girders next.
For the webs of I-girders:
• D/t w ≤ 150 with no longitudinal stiffeners
• D/t w ≤ 300 with longitudinal stiffeners
For the flanges of I-girders:
•
•
•
•
•
•
flange width (bf )/thickness (tf ) limit: bf ≤ 24 tf
flange width/web depth limit: bf ≥ D/6
flange thickness/web thickness limit: tf ≥ 1.1 t w
0.10 ≤ Iyc/Iyt ≤ 10
Iyc = (tf )(bf )3/12, compression flange
Iyt = (tf )(bf )3/12, tension flange
Stiffened webs must satisfy Equation 6.4, or there is a penalty on shear resistance.
Subscripts “c” and “t” refer to the compression and tension flanges, respectively.
2 Dt w
£ 2.5
b fct fc + b ft t ft
(6.4)
6.1 FLEXURAL RESISTANCE AT THE STRENGTH LIMIT STATE
For complete coverage of flexural resistance at each limit state for all types of steel
girders, refer to the AASHTO LRFD BDS, Section 6.10 for I-girders and Section
6.11 for box girders. Discussions in this chapter are limited to (a) positive flexural
resistance of composite, compact sections at the Strength limit state, Section 6.1.1,
(b) positive flexural resistance of non-compact composite sections at the Strength
limit state in Section 6.1.2, and (c) negative flexure at the Strength limit state in
Section 6.1.3.
6.1.1 Composite Compact Sections in Positive Flexure
Nominal, positive flexural resistance of compact sections is determined by Equation
6.5a. Dp is the distance from the extreme compression fiber to the plastic neutral axis
(PNA). Dt is the total composite girder depth, including the deck. In continuous span
homogeneous girders, the moment resistance, Mn, is further limited to 1.3 × My, as
indicated in Equation 6.5a. To find My, solve Equation 6.5b for MAD, using factored
moments MD1 and MD2. MD1 is the factored dead load moment carried by the girder
alone (NC denotes “non-composite”). MD2 is the factored dead load moment carried by the long-term composite section (LT denotes “long-term composite”). Then
substitute into Equation 6.5c. Refer to the AASHTO LRFD BDS, Section D6.2, for
more information on the computation of My. Figure 6.1 will clarify the definitions
of the various depths for a composite section in positive flexure (top of the deck in
compression).
127
Steel Welded Plate I-Girders
FIGURE 6.1
Composite section in positive flexure.
Refer to Section 6.9 of this chapter for a more detailed discussion on the computation of plastic moment resistance, Mp, for non-composite and composite steel
I-girders.
ì
ïM p,
ï
ï æ
Dp ö
,
ï M p ç 1.07 - 0.7
Dt ÷ø
Mn = í è
ï
ï0,
ï
ï£ 1.3R M ( continuous spans )
h
y
î
if
Dp
£ 0.1
Dt
if 0.10 <
if
Dp
£ 0.42
Dt
(6.5a)
Dp
> 0.42 ( not permitted )
Dt
M D1 M D 2 M AD
+
+
S NC
SLT
SST
(6.5b)
M y = M D1 + M D 2 + M AD
(6.5c)
Fyf =
Mu +
1
fl S xt £ f f M n
3
(6.5d)
The required design resistance and available design resistance must satisfy Equation
6.5d. The lateral flange bending stress, f l, may be determined according to Equation
6.9 and as discussed below. The section modulus, Sxt, is that for the tension flange
and is equal to Myt/Fyt. Refer to AASHTO LRFD BDS, Article D6.2, for additional
information on the computation of Myt.
Positive moment sections which are composite in the final conditions require
assessment at various stages of construction. Significant loading due to wet concrete
and construction live loads are a critical consideration. Stability assessment prior to
the deck reaching design strength is particularly critical. For the completed structure,
the deck provides lateral bracing of the top flange under typical circumstances. Prior
128
LRFD Bridge Design
to the deck reaching design strength, the top flange is more susceptible to lateraltorsional buckling. Refer to Section 6.7 of this chapter for a more detailed discussion
of stability bracing requirements.
6.1.2 Non-Compact Composite Sections in Positive Flexure
Non-compact, positive moment, composite sections must be assessed for the Strength
limit state based on stress analysis for the various stages of the construction process,
namely non-composite (NC), long-term composite (LT), and short-term composite
(ST) properties. Equation 6.6 is the basic stress equation for flexure due to loads at
the various stages of the construction process. The stress, f bu, is computed without
contribution from lateral flange bending.
For non-compact, positive moment, composite sections, Equation 6.7 must be
satisfied for the compression flange and Equation 6.8 must be satisfied for the tension flange, both under Strength limit state load factors. The first-order lateral flange
bending stress, f l, may be estimated using Equation 6.9, with consistent units, for
straight I-girders under wind-induced moment, Mw. The compression flange nominal
resistance, Fnc, and the tension flange nominal resistance, Fnt, are given by Equation
6.10 and 6.11, respectively. For homogeneous (non-hybrid) girders, the hybrid factor,
Rh, is equal to 1.00. For hybrid girders, refer to Section 6.10.1.10.1 of the AASHTO
LRFD BDS. The web load-shedding factor, Rb, is also equal to 1.00 for composite
sections in positive flexure, with D/t no more than 150. For other cases, refer to
Section 6.10.1.10.2 of the AASHTO LRFD BDS.
æM
fbu = ± ç NC
è S NC
ö æ M LT
÷±ç S
ø è LT
ö æ M ST ö
÷±ç S ÷
ø è ST ø
fbu £ f f Fnc
fbu +
(6.6)
(6.7)
1
fl £ f f Fnt
3
(6.8)
6 Mw
tb2
(6.9)
fl =
Fnc = Rb Rh Fyc
(6.10)
Fnt = Rh Fyt
(6.11)
For composite deck bridges, the wind load on the lower half of the outside I-girder
may be assumed to be carried by the bottom flange in lateral bending. The wind
load on the upper half of the I-girder may be assumed to be transmitted directly to
the concrete deck. Frame action of cross-frames may be used, essentially treating
the wind-loaded flange as a continuous beam supported at cross-frame locations. As
129
Steel Welded Plate I-Girders
a result, the wind moment, Mw, may be taken to be equal to that given by Equation
6.12. The load, W, is the wind load per foot-length acting on the flange, and may be
calculated using Equation 6.13 for the composite deck conditions previously stated.
The girder depth is d, and PD is the design wind pressure at the appropriate Strength
limit state. The cross-frame spacing is Lb.
WL2b
10
(6.12)
hig PD d
2
(6.13)
Mw =
W=
Flange lateral bending stress, f l, may be taken to be equal to the first-order value
from Equation 6.9, provided Equation 6.14 is satisfied. Otherwise, the value calculated from Equation 6.9 requires amplification to account for second-order effects.
Refer to the AASHTO LRFD BDS, Section 6.10.1.6. See also AASHTO LRFD
BDS, Section 6.10.8.2.3, for a detailed calculation of the moment gradient factor, Cb,
in cases where a value other than Cb = 1.00 may be used.
Lb £ 1.2 L p
Cb Rb
fbu / Fyc
(6.14)
Factored lateral flange bending stress, f l, at the Strength limit state must never exceed
0.6Fyf.
For computation of lateral flange bending stresses in curved girders, the reader is
referred to Chapter 6 of the AASHTO LRFD BDS (AASHTO, 2020).
6.1.3 Negative Flexure and Non-composite Sections
For sections in negative flexure and for non-composite sections, Equation 6.15a
defines the compression flange resistance to local buckling. Equation 6.15b defines
the compression flange resistance to lateral torsional buckling. The web load-shedding factor, Rb, is given by Equation 6.24b for cross sections in negative flexure and
non-composite cross sections. Fnc is never to be taken to be greater than RbRhFyc.
The flange stresses at the Strength limit state must satisfy Equations 6.15g (for the
compression flange) and 6.15h (for the tension flange). Alternative methods from
Appendix A6 of the AASHTO LRFD BDS may be followed in lieu of those discussed here for non-composite sections and sections in negative flexure. Further
limitations on sections in negative flexure and non-composite sections are given in
Equations 6.20 and 6.21. The moment gradient factor, Cb, may conservatively be
taken to be equal to 1.0, representing an unbraced segment with constant moment
over the entire segment. Refer to the AASHTO LRFD BDS, Article 6.10.8.2.3, for
detailed calculation of Cb to justify values greater than 1.0.
Equation 6.21 for web slenderness must be satisfied in order to permit the alternative Appendix A6 method for evaluating the cross section. Otherwise, the stress
analysis and limits presented here for non-compact sections are applicable.
130
LRFD Bridge Design
ì Rb Rh Fyc ,
ï
Fnc = í é æ
Fyr ö æ l f - l pf ö ù
ï ê1 - ç 1 - R F ÷ ç l - l ÷ ú Rb Rh Fyc ,
h yc ø è rf
pf ø ú
û
î êë è
ì Rb Rh Fyc ,
ï
F ö æ L - L p öù
ï é æ
Fnc = íCb ê1 - ç 1 - yr ÷ ç b
÷ ú Rb Rh Fyc ,
ï ëê è Rh Fyc ø è Lr - L p ø úû
ïF £ R R F ,
b h yc
î cr
E
Fyc
L p = 1.0rt
Lr = p rt
rt =
E
Fyr
b fc
æ 1 Dct w ö
12 ç 1 +
÷
è 3 b fct fc ø
Fcr =
Cb Rbp 2 E
( r)
Lb
2
if l f £ l pf
if l f > l pf
(6.15a)
if Lb £ L p
if L p < Lb < Lr
(6.15b)
if Lb > Lr
(6.15c)
(6.15d)
(6.15e)
(6.15f)
t
1
fl £ f f Fnc
3
(6.15g)
1
fl £ f f Rh Fyt
3
(6.15h)
b fc
2t fc
(6.16)
fbu +
fbu +
lf =
l pf = 0.38
E
Fyc
(6.17)
lrf = 0.56
E
Fyc
(6.18)
131
Steel Welded Plate I-Girders
Fyr = 0.70 Fyc , for homogeneous girders
(6.19)
Fyf £ 70 ksi
(6.20)
2Dc
£ lrw
tw
(6.21)
I yc
³ 0.30
I yt
(6.22)
lrw = 4.6
æ
E
5.0 ö E
E
£ 3.1 +
£ 5.7
Fyc çè
awc ÷ø Fyc
Fyc
awc =
Rb = 1.0 •
•
•
•
•
•
•
•
•
•
2 Dct w
b fct fc
æ 2 Dc
ö
awc
- lrw ÷
ç
1200 + 300awc è t w
ø
(6.23)
(6.24a)
(6.24b)
Dc = depth of web in compression in the elastic stress range, inches
bfc = compression flange width, inches
tfc = compression flange thickness, inches
bft = tension flange width, inches
tft = tension flange thickness, inches
Fyc = compression flange yield stress, ksi
t w = web thickness, inches
d = total girder depth, inches
Iyc = tfc(bfc)3/12
Iyt = tft(bft)3/12
The material presented here is applicable for straight I-girder bridges, with no skew,
and with cross-frame lines continuous across the cross section of the bridge.
At locations with holes in the tension flange (at field splices, for example), for the
Strength limit state and for constructability, the computed flange stress is limited to
0.84(An/Ag)Fu ≤ Fy. Refer to AASHTO LRFD BDS, Section 6.10.1.8, for a detailed
discussion.
6.2 SHEAR RESISTANCE
The shear resistance of stiffened I-girders is given by Equation 6.25. The shear resistance of unstiffened panels is computed using the equation for no tension field action
(TFA) and k = 5. The resistance factor for shear in steel I-girders is ϕv = 1.00. The
clear distance between transverse stiffeners is do. The plastic shear, VP = 0.58FywDt w.
132
LRFD Bridge Design
CVp for end panels with do £ 1.5D, No TFA
ì
ï
ù
ï é
ï ê
ú
Vn = í
êC + 0.87 (1 - C ) ú for intermediate panels, TFA
V
p
ï ê
2 ú
ï ê
1 + do
D úû
ïî ë
(6.25)
( )
k = 5+
ì
ï
ï1
ï
ï
ï 1.12 Ek
C=í
ï D t w Fyw
ï
ï 1.557 Ek
×
ï
2
Fyw
ï D
î tw
( )
if
5
( do D )
(6.26)
D
Ek
£ 1.12
tw
Fyw
Ek
D
Ek
£ £ 1.40
Fyw tw
Fyw
if 1.12
if
2
(6.27)
D
Ek
³ 1.40
tw
Fyw
These equations are applicable to transversely stiffened I-girders. The design requirements for the transverse stiffeners will be presented next. The design requirements
for I-girders with longitudinal stiffeners is beyond the scope of this text.
6.3 TRANSVERSE STIFFENER DESIGN
Stiffeners in straight girders not used as connection plates should be tightly fitted or
attached at the compression flange but need not be in bearing with the tension flange.
Single-sided stiffeners on horizontally curved girders should be attached to both
flanges. When pairs of transverse stiffeners are used on horizontally curved girders,
they must be fitted tightly or attached to both flanges.
Stiffeners used as connecting plates for diaphragms or cross-frames must be
attached to both flanges.
The distance between the end of the web-to-stiffener weld and the near edge of
the adjacent web-to-flange weld must be no less than 4t w but cannot exceed the lesser
of 6t w or 4.0 inches.
Transverse stiffener dimensions must satisfy Equations 6.28 and 6.29. Properties
of transverse stiffeners must satisfy Equation 6.30.
bt ³ 2 + D
16t p ³ bt ³
30
bf
4
(6.28)
(6.29)
133
Steel Welded Plate I-Girders
1.5
ì
é
D 4 rt1.3 æ Fyw ö ù
ïMin êbt w3 J ,
ú , with no TFA
40 çè E ÷ø ú
ï
êë
û
It ³ í
1.5
4
1
.
3
ï
D rt æ Fyw ö
, with TFA
ï
40 çè E ÷ø
î
J=
2.5
( D)
do
Fcrs =
2
- 2.0 ³ 0.50
0.31E
(6.30)
(6.31)
£ Fys
(6.32)
ü
ìF
rt = Max í yw ,1.0 ý
F
þ
î crs
(6.33)
b = Min {do , D}
(6.34)
æ bt ö
ç tp ÷
è
ø
2
• It = moment of inertia of the transverse stiffener taken about the edge in
contact with the web for single stiffeners and about the mid-thickness of the
web for stiffener pairs, in4.
• do = the smaller spacing on either side of the stiffener, inches
• bt = stiffener width, inches
• tp = stiffener thickness, inches
• Fys = Fy for the stiffener, ksi
6.4 BEARING STIFFENER DESIGN
Bearing stiffeners are placed on the webs of built-up I-girder sections at all bearing
locations. Bearing stiffeners are comprised of plates or angles welded or bolted to
both sides of the web, must extend the full depth of the web, and extend as closely
as practical to the outer flange edges. Each bearing stiffener must be finished to
bear against the flange through which load is transmitted. This would be the bottom
flange in a continuous I-girder at intermediate piers. Bearing stiffeners serving as
connection plates must be attached to both flanges of the cross section as specified in
AASHTO LRFD BDS, Section 6.6.1.3.1.
Bearing stiffeners are checked for proportion limits, bearing resistance, and axial
resistance. For the bearing resistance check, only the portion of the clipped stiffener
in contact with the flanges is included as the effective area. For the axial resistance
check, the full stiffener width may be used along with a strip of web extending no
more than 9t w on each side of the stiffeners to form a column of effective length
KL = 0.75D.
134
LRFD Bridge Design
Equation 6.35 provides the proportion check criterion for bearing stiffeners.
Equation 6.36 gives the bearing resistance. Axial resistance is determined using
Equation 6.37. “r” is the radius of gyration of the effective column section about the
mid-thickness of the web. Bearing resistance and axial resistance must both exceed
the Strength limit state reaction.
E
Fys
bt £ 0.48t p
(6.35)
fb ( Rsb )n = 1.0 (1.4 Apn Fys )
(6.36)
fc Pn = 0.95Pn
(6.37)
Pe =
(
p 2E
KL
r
)
2
Ag
(6.38)
Po = Fy Ag
(6.39)
Po
ìé
ù
P
ï ê0.658 Pe ú Po , if e ³ 0.44
Po
ï
úû
Pn = í êë
ï
P
if e < 0.44
ï 0.877Pe ,
Po
î
(6.40)
6.5 FATIGUE DESIGN
Particularly with steel girder bridges, it is critical to consider Fatigue in design.
Fatigue is of concern any time cyclic stress exists, with one extreme of the stress
range being tension. Fatigue may be described as the (a) initiation and (b) propagation of cracks at stress levels which may be well below yield.
Repeated application of cyclic stress, in terms of stress range, may produce extensive damage in structural elements. Figure 6.2 depicts multiple Fatigue resistance
curves defining the number of cycles to failure for various stress ranges and detail
categories.
Bridge elements and details are assigned a Fatigue Category. These are represented by the various curves in Figure 6.2. Each Fatigue Category has associated
parameters, A and ∆FTH. The various categories in AASHTO LRFD BDS and the
associated parameters are summarized in Table 6.1. For those familiar with Fatigue
requirements for building design in steel, there are differences between AASHTO
for bridges and AISC 360-16 for buildings, and the requirements from the different
specifications must not be confused.
Table 6.2 is a small sample of the many details with assigned Fatigue Category
found in AASHTO LRFD BDS (AASHTO, 2020).
135
Steel Welded Plate I-Girders
FIGURE 6.2 Typical fatigue curve.
TABLE 6.1
Fatigue Parameters
Category
ΔFTH, ksi
A, ksi3
A
250 × 10
8
24.0
B
B’
C
C’
D
E
120 × 108
61 × 108
44 × 108
44 × 108
22 × 108
11 × 108
16.0
12.0
10.0
12.0
7.0
4.5
E’
3.9 × 108
2.6
AASHTO considers two distinct Fatigue limit states:
• Fatigue I, infinite life
• Fatigue II, finite life
For each Fatigue limit state, Equation 6.41 defines the criteria used to assess fatigue
resistance for any element, based on actual stress range (∆f) and permissible stress
range (∆F).
136
LRFD Bridge Design
TABLE 6.2
Sample Fatigue Category Assignments
Brief Descriptiona
Category
Base metal, except uncoated weathering steel
A
Base metal, uncoated weathering steel
Base metal at re-entrant corners, copes, cuts, …
Base metal at the net section of open holes
Base metal and weld metal in web-to-flange continuous fillet welded plate girders
Base metal at ends of partial length welded cover-plates narrower than the flange
Base metal at the toe of stiffener-to-flange and stiffener-to-web fillet welds
Base metal at steel headed stud anchors
a
B
C
D
B
E or E’
C’
C
See AASHTO LRFD BDS (AASHTO, 2020) for complete descriptions
The load factor, γ, and the limiting stress range, (∆F)n, are as follows:
• For Fatigue I (infinite life), γ = 1.75 with (∆F)n from Equation 6.41b
• For Fatigue II (finite life), γ = 0.80 with (∆F)n from Equation 6.41c
The actual stress range, ∆f, is computed using the fatigue truck with an impact factor, IM, = 0.15, without application of the multi-presence factor (m), and a single-lane
live load distribution factor.
The number of cycles, N, expected over the life of the bridge is dependent upon
(a) the anticipated 75-year, single-lane, average daily truck traffic (ADTTSL), and
(b) the number of stress cycles per truck passage, n. The ADTTSL would ideally be
based on reliable projections. When such projections are not available, Section 3.6.1
of AASHTO LRFD BDS provides means of estimating the ADTTSL when either
the average daily traffic (ADT) or the average daily truck traffic (ADTT) is known.
Equation 6.41d may be used in such cases. Calculation of the total number of cycles
follows from Equation 6.41e.
g ( Df ) £ ( DF )n
(6.41a)
( DF )n = ( DF )TH
(6.41b)
( DF )n
æ Aö
=ç ÷
èNø
1
3
(6.41c)
ADTTSL = p ´ ADTT = fr ´ p ´ ADT
(6.41d)
N = 365 ´ 75 ´ n ´ ADTTSL
(6.41e)
Steel Welded Plate I-Girders
137
The fraction of trucks in traffic, fr, may be estimated as follows:
•
•
•
•
Rural interstate, fr = 0.20
Urban interstate, fr = 0.15
Other rural, fr = 0.15
Other urban, fr = 0.10
The fraction of truck traffic in a single lane may be estimated from:
• 1 lane available to trucks, p = 1.00
• 2 lanes available to trucks, p = 0.85
• 3 or more lanes available to trucks, p = 0.80
The number of stress cycles per truck passage, n, is taken to be equal to 1.5 for a
continuous, longitudinal I-girder at regions near interior supports, and 1.0 elsewhere
for the I-girder.
6.6 FIELD SPLICE DESIGN
The maximum length which may be shipped to a site varies depending on the quality and geometry of roads, availability of equipment, etc. Usually, 140 ft to 160 ft is
a typical maximum length which may be shipped. For steel girder bridges, the field
sections are shipped to the site and bolted together in place. Figure 6.3 depicts an
example of a field splice detail.
Flange splice filler plates are required any time the flange thickness is not the
same for the two sections on opposing sides of the splice centerline.
The National Steel Bridge Alliance (NSBA) provides a free Excel spreadsheet
for field splice analysis and design, NSBA-Splice (https://www​.aisc​.org​/nsba​/design​resources​/nsba​-splice/).
FIGURE 6.3
Steel welded plate girder field Splice.
138
LRFD Bridge Design
For extensive design examples by hand calculation, refer to Bolted Field Splices
for Steel Bridge Flexural Members: Overview and Design Examples (Grubb et al.,
2018), also published by the NSBA.
Splice elements include:
•
•
•
•
•
flange splice plates (inner and outer)
flange bolts
web splice plates
web bolts
filler plates
Bolted field splices are to be designed slip-critical at the Service II limit state.
The flange splice design force, Pfy, at the Strength limit state is determined according to Equations 6.42 through 6.44. The controlling flange force for each flange is the
smaller of the two values on each side of the splice at said flange. In the 9th edition of
AASHTO LRFD BDS, ϕu = 0.80 and ϕy = 0.95.
Pfy = Fyf × Ae
(6.42)
fu Fu
× An £ Ag
fy Fy
(6.43)
An = t f éëb f - ndh ùû
(6.44)
Ae =
For standard holes, the diameter of the holes, dh, is taken to be the bolt diameter plus
1/16 inches for bolt diameters equal to 7/8 inches or less. For larger bolts, the diameter of the holes, dh, is taken to be equal to the bolt diameter plus 1/8 inch. Unlike
the requirements from AISC 360-16 for buildings, no deduction for hole damage is
incorporated into AASHTO requirements for net area calculation.
For a flange splice with inner and outer splice plates, Pfy at the Strength limit state
may be assumed to be divided equally between the inner and outer plates and their
connections when the areas of the inner and outer plates do not differ by more than
ten percent. In this case, the connections are proportioned assuming double shear.
Should the areas of the inner and outer plates differ by more than ten percent, the
design force in each splice plate and its connection at the Strength limit state should
instead be determined by multiplying Pfy by the ratio of the area of the splice plate
under consideration to the total area of the inner and outer splice plates. For this case,
the connections are proportioned for the maximum calculated splice-plate force acting on a single shear plane.
The determination of web splice design forces is a bit more complicated. The
vertical component of web splice design force, Vr, is taken to be equal to the shear
resistance of the web, ϕVn, at the point of splice (see Section 6.2 of this chapter).
The Strength limit state moment at the splice location is compared to the moment
resistance provided by the flanges. If the Strength limit state moment is greater than
the flexural resistance of the flanges, then an additional horizontal force, Hw, must be
139
Steel Welded Plate I-Girders
included in the design to enable the web to carry the additional moment. It is, therefore, necessary to establish a means of determining the moment resistance provided
by the flanges.
For composite sections in positive bending (compression in the deck), a moment
arm, A, is taken to be equal to the distance between the centroids of the bottom
flange and the concrete deck. The flexural resistance provided by the flanges, Mfl,
is then taken to be equal to the moment arm, A, times the bottom flange design
force, Pfy.
The moment arm, A, for sections in negative flexure and for non-composite sections, is taken to be equal to the distance between the top and bottom flange centroids. The force to be used in calculating the flange moment is the smaller of Pfy for
the top flange splice and Pfy for the bottom flange splice.
Should the Strength limit state moment, MuStr, be larger than the moment resistance provided by the flanges, then the additional horizontal web splice force, Hw,
is determined from Equation 6.45. For composite sections in positive flexure, Aw is
equal to the distance between the mid-height of the web and the deck centroid. For
non-composite sections and sections in negative flexure, Aw is equal to one-quarter
of the web depth. The horizontal and vertical components of web splice force are
combined in a vector fashion shown in Equation 6.46.
MuStr - M fl
Aw
(6.45)
Rweb = Vr2 + H w2
(6.46)
Hw =
For bolt shear resistance at the Strength limit state, the resistance factor, ϕ, is equal
to 0.80, and Equation 6.47 or 6.48, as appropriate, provides the nominal resistance.
Rn = 0.56 Ab Fub N s ( threads excluded from shear planes )
(6.47)
Rn = 0.45 Ab Fub N s ( threads not excluded from shear planes )
(6.48)
The bolt tensile strength, Fub, is 120 ksi for ASTM F3125, Grade A325 bolts,
and 150 ksi for Grade A490 bolts. Ab is the bolt area corresponding to the nominal bolt diameter and Ns is the number of shear planes. If the distance between
extreme bolts is greater than 38 inches, a penalty factor of 0.83 is to be applied to
the bolt shear resistance, R n. If filler plates 0.25 inches in thickness or greater are
used, an additional penalty factor is to be applied to R n. The filler plate penalty
factor, R, is given by Equations 6.49 and 6.50. Af is the area of the filler plate and
Ap is the smaller of either (a) the flange area or (b) the sum of the splice plate
areas.
R=
1+ g
1 + 2g
(6.49)
140
LRFD Bridge Design
g =
Af
Ap
(6.50)
Block shear in splice plates and flange plates must be checked. Equation 6.51 provides the nominal resistance. The design resistance, ϕRn, incorporates a resistance
factor, ϕ, = 0.80, for block shear checks. Ubs is equal to 1.0 for splice plates in tension.
Rp is 1.0 for splice plates since holes punched full size are not permitted in splices.
Rn = R p ( 0.58Fu Avn + U bs Fu Atn ) £ R p ( 0.58Fy Avg + U bs Fu Atn )
(6.51)
0.90 for bolt holes punched full size
ì
ï
Rp = í
1.0 for bolt holes drilled fulll size
ï1.0 for bolt holes sub-punched and reamed to size
î
(6.52)
Resistance factors for additional design checks include ϕ = 0.80 net section rupture
and ϕ = 0.95 for gross section yielding of flange plates in tension. Nominal resistances for these limit states are given in Equations 6.53 and 6.54.
Rn = Fy Ag for yielding
(6.53)
Rn = Fu An R pU for rupture
(6.54)
The shear lag factor, U, is equal to 1.0 for splice plates in tension.
For web splice plates in shear, Equations 6.55 for yielding and 6.56 for rupture
are the applicable checks. The resistance factor for shear yielding is ϕ = 1.0, and that
for shear rupture is ϕ = 0.80.
Rn = 0.58Fy Agv for yielding
(6.55)
Rn = 0.58Fu Avn R p for rupture
(6.56)
Web bolts should be checked for slip under the Service II limit state factored shear
at the splice, or the deck casting shear (multiplied by a typical load factor of 1.4),
whichever is larger. If the flange bolts are not able to resist slip for the factored
moment, then the web bolts will require additional slip checks. See the literature
(Grubb et al., 2018) for details on such checks.
The combined areas of the flange splice plates must equal or exceed the area of
the smaller flanges to which they are attached. Web plate areas must equal or exceed
the web area.
Fatigue generally need not be checked for bolted field splice designs because the
current version of the specification provisions is designed to preclude Fatigue failure
in the splice. Refer to the AASHTO LRFD BDS commentary in Section 6.13.6.1.3
for further discussion.
Slip-critical provisions are to be met at the Service II limit state for the completed
structure on the composite section. Slip-critical provisions are also to be satisfied on
141
Steel Welded Plate I-Girders
the non-composite section for loads during casting of the deck, with a typical load
factor of 1.4 applied to the maximum deck casting moment. Slip resistance, Rn, is
computed as shown in Equation 6.57.
Rn = K s K h N s Pt
(6.57)
The required minimum bolt tension, Pt, the hole-related factor, K h, and the surface-related factor, K s, are given in Tables 6.3 through 6​ .5. Ns is the number of
slip surfaces, typically equal to 2 for steel bridge girder field splices as long as
the area on inner-to-outer splice plates is within 0.90 to 1.10 so that the flange
splice force may be assumed to be equally distributed between inner and outer
splice plates.
TABLE 6.3
Minimum Bolt Pre-tension, Pt (kips)
Bolt Diameter
Grade A325 Bolts
Grade A490 Bolts
⅝ inch
19
24
¾ inch
⅞ inch
1 inch
1 ⅛ inch
1 ¼ inch
1 ⅜ inch
28
39
51
64
81
97
35
49
64
80
102
121
1 ½ inch
118
148
TABLE 6.4
Slip Factor Kh
Standard holes
1.00
Over-sized and short-slotted holes
Long-slotted holes, slot perpendicular to force
0.85
0.70
Long-slotted holes, slot parallel to force
0.60
TABLE 6.5
Slip Factor Ks
Class A surface conditions
0.30
Class B surface conditions
Class C surface conditions
0.50
0.30
Class D surface conditions
0.45
142
LRFD Bridge Design
Class A surface conditions refer to “unpainted clean mill scale, and blastcleaned surfaces with Class A coatings” (AASHTO, 2020).
Class B surface conditions refer to “unpainted blast-cleaned surfaces to
SSPC-SP 6 or better, and blast-cleaned surfaces with Class B coatings, or
unsealed pure zinc or 85/15 zinc/aluminum thermal-sprayed coatings with
a thickness less than or equal to 16 mils” (AASHTO, 2020).
Class C surface conditions refer to “hot-dip galvanized surfaces” (AASHTO, 2020).
Class D surface conditions refer to “blast-cleaned surfaces with Class D
coatings” (AASHTO, 2020).
Bolt limitations include:
• center–center spacing ≥ 3d
• center–center space on a free edge ≤ 4 + 4t ≤ 7 inches (t = thickness of thinner part)
• oversized and slotted holes not to be used
• no fewer than 2 rows of bolts on each side of the splice centerline
• minimum bolt size is ¾-inch diameter
Other checks to be made for splice bolts and splice plates include:
•
•
•
•
•
•
bolt shear
bearing on bolt holes
slip of connected plates
block shear
gross section yielding
net section fracture
6.7 STABILITY BRACING
The AASHTO LRFD BDS (AASHTO, 2020) does not outline stability bracing
requirements in detail. Excellent guidance is available in Volume 13 of the Federal
Highway Administration (FHWA) Steel Bridge Design Handbook (FHWA, 2015).
Another valuable source of relevant information may be found in a recent National
Cooperative Highway Research Program Report (National Academies of Sciences,
Engineering, and Medicine, 2021). The material presented here is based largely on
these documents along with American Institute of Steel Construction, AISC 360-16
(AISC, 2016). Lateral bracing of the compression flange in bridge girders is panel
bracing, according to AISC 360-16, Appendix 6 terminology.
The required shear strength and shear stiffness of panel bracing are given by
Equations 6.58 and 6.59, respectively (AISC Appendix 6, Equations A.6.5 and
A.6.6a):
æMC ö
Vbr = 0.01 ç u d ÷
è ho ø
(6.58)
143
Steel Welded Plate I-Girders
b br ³
1 æ 4 MuCd ö
×
f çè Lbr ho ÷ø
(6.59)
• ϕ = resistance factor = 0.75
• Mu = required LRFD flexural strength within the panel being considered,
inch-kips
• Cd = 2.0 for the brace closest to the inflection point in a beam subject to
double curvature bending
• Cd = 1.0 for all other cases
• Lbr = unbraced length of the panel under consideration, inches
• ho = distance between flange centroids, inches
The actual stiffness of the panel bracing may be estimated, assuming that the girder
area is much larger than the brace area (typically a reasonable assumption), resulting
in Equation 6.60.
æ E ö æ 4 A1 A2 cos3 q ö
b br = ç ÷ ç
÷
3
è S ø è 4 A1 + A2 cos q ø
(6.60)
Note that Mu /ho is an estimate of the flange force. Conservatively, one could take Mu /
ho equal to Fy for the flange times the area of the flange.
A laterally braced panel is shown in Figure 6.4 in plan view.
Cross-frames and diaphragms are torsional bracing systems. Until AASHTO
incorporates requirements for such systems into the AASHTO LFRD BDS, AISC
Appendix 6 Equation A.6.9 gives the required strength of the torsional brace,
repeated here in Equation 6.61a. A recent study (National Academies of Sciences,
Engineering, and Medicine, 2021) concluded that this may provide unsatisfactory
results, and Equation 6.61b is the recommended torsional brace strength from that
study.
M br = 0.02 Mu
FIGURE 6.4 Lateral (panel) bracing of a bridge girder flange.
(6.61a)
144
LRFD Bridge Design
M br =
0.036 Mu L
nCb Lb
(6.61b)
Actual system stiffness, βT, is composed of three springs in series and is given by
Equation 6.62.
• brace stiffness, βb
• web distortion stiffness, βsec
• girder system stiffness, βg
1
1
1
1
=
+
+
bT b b bsec b g
(6.62)
Equation 6.63 provides the required system stiffness (from AISC Appendix 6,
Equation A.6.11a). However, in accord with the National Academies report (National
Academies of Sciences, Engineering, and Medicine, 2021), the coefficient 2.4 has
been replaced with the coefficient 3.6 in the given equation, so there is a 50% difference between the equation shown in AISC and that given here.
1 3.6 L
bT ³ ×
f nEI yeff
æM ö
×ç u ÷
è Cb ø
2
(6.63)
• E = modulus of elasticity for steel, 29,000 ksi
• Iyeff = effective out-of-plane moment of inertia of the beam being braced, in4
t
I yeff = I yc + I yt
c
(6.64)
•
•
•
•
•
•
•
Iyc = moment of inertia of compression flange about y-axis, in4
Iyt = moment of inertia of tension flange about y-axis, in4
t = distance from neutral axis to extreme tension fiber, inches
c = distance from neutral axis to extreme compression fiber, inches
L = length of span being braced, inches
ϕ = resistance factor = 0.75
Mu = largest factored moment in unbraced segment, containing the brace, of
the span being braced, inch-kips
• Cb = LTB (lateral torsional buckling) modification factor
• n = number of brace points within the span being braced
The web distortional stiffness is given by Equation 6.65 (Equation A.6.12 of AISC,
Appendix 6).
bsec =
3.3E æ 1.5hot w3 t st bs3 ö
+
ç
÷
ho è 12
12 ø
(6.65)
145
Steel Welded Plate I-Girders
•
•
•
•
ho = distance between flange centroids of the beam being braced, inches
t w = web thickness of the beam being braced, inches
tst = thickness of web stiffeners on the beam being braced, inches
bs = stiffener width for one-sided stiffeners; twice the individual stiffener
width for pairs of stiffeners, inches
The stiffness of the torsional brace component of the system is provided by Equations
6.66 through 6.69 for various configurations (Yura, 2001).
bb =
bb =
bb =
bb =
6 EI b
for diaphragms
S
2 ES 2hb2
for K -frame cross-frames
8L3c S 3
+
Ac Ah
ES 2hb2
for Z -frame cross-frames
2 L3c S 3
+
Ac
Ah
Ac ES 2hb2
for X -frame cross-frames
L3c
(6.66)
(6.67)
(6.68)
(6.69)
The parameters used in the stiffness equations are as follows:
•
•
•
•
•
•
S = girder spacing, inches
Ah = area of horizontal cross-frame member, in2
Ac = area of diagonal cross-frame member, in2
Lc = length of diagonal cross-frame member, inches
hb = height of cross-frame, inches
Ib = moment of inertia of diaphragm member, in4
The equation for diaphragm stiffness (Equation 6.66), is based on the assumption
that the diaphragm is located near the compression flange, forcing the diaphragm
into double curvature bending under girder buckling. Should diaphragms be located
near the tension flanges, as may be the case with railway through-girders and similar structures, then the equation must be modified to incorporate a coefficient of 2,
rather than 6.
The girder system stiffness, βg, is given by Equation 6.70.
bg =
24 ( ng - 1)
ng
2
×
S 2 EI x
L3
(6.70)
146
•
•
•
•
LRFD Bridge Design
Ix = strong axis moment of inertia for one girder of the girders being braced, in4
ng = number of girders being braced
S = spacing of the girders being braced, inches
L = length of span being braced, inches
Torsional bracing may become ineffective in systems for which Lg /S becomes
very large. The buckling capacity of a single girder of a twin-girder system in the
global system buckling mode may be estimated from Equation 6.71 (Helwig and
Yura, 2015).
M gs =
p 2 SE
I yeff I x
L2g
(6.71)
For a 3-girder system, Equation 6.71 may be used by replacing Iyeff with 1.5Iyeff and
S with 2S.
For a 4-girder system, Equation 6.71 may be used by replacing Iyeff with 2Iyeff and
S with 3S.
One method of increasing stability is through the use of combined lateral braces
and torsional braces. End-panel bracing up to b = 0.20L has been shown to be
effective.
6.8 SHEAR STUDS
Shear studs welded to the top flange of steel I-girders provide for composite action
between the girder and the bridge deck.
The ratio of the height to the diameter of the shear must be no less than 4.0.
The center-to-center spacing of studs must not exceed 48 inches for members having a web depth greater than or equal to 24 inches. For members with a web depth
less than 24 inches, the center-to-center spacing must not exceed 24 inches. The
center-to-center spacing of studs is also limited to no less than six stud diameters.
Stud shear connectors must be no closer than 4.0 stud diameters center-to-center
transverse to the longitudinal axis of the I-girder. The clear distance between the
edge of the top flange and the edge of the nearest stud must not be less than 1 inch.
The clear depth of concrete cover over the top of studs must be at least 2 inches.
Studs must extend at least 2 inches into the concrete deck.
Shear stud requirements include those for the Fatigue I limit state and for the
Strength limit state.
Fatigue limit state requirements are summarized in Equations 6.72 through 6.76.
Zr is the resistance for a single stud. The required pitch (longitudinal spacing) of
studs for the Fatigue limit state is p. The factor, w, is equal to 24 inches at end supports and 48 inches otherwise. Frc is the range of cross-frame forces at the top flange
for the Fatigue limit state. Vf is the vertical shear range due to Fatigue limit state
loading. The number of shear connectors transverse to the girder centerline is n per
row. The equation for Ffat is valid only for straight girders. QST and IST are properties of the short-term composite section, calculated using a modular ratio, n = ES/EC
147
Steel Welded Plate I-Girders
(not to be confused with the number of studs per row, n, appearing in Equation 6.73;
context should reveal which n-parameter is under discussion).
Z r = 5.5d 2
nZ r
Vsr
p£
Vsr =
(V fat ) + ( Ffat )
2
V fat =
(6.72)
(6.73)
2
(6.74)
V f QST
I ST
(6.75)
Frc
w
(6.76)
Ffat =
Strength limit state requirements for shear studs are summarized in Equations
6.77 through 6.84. The minimum yield and tensile strengths of studs are specified
in the AASHTO LRFD BDS, Section 6.4.4, to be F y = 50 ksi and Fu = 60 ksi. For
Strength limit state criteria, n is the number of total shear studs required between
two locations under consideration. The cross-sectional area, Asc, of a single shear
stud is simply πd2/4. For straight girders, both FP and F T may be taken to be
equal to zero. For curved girders, the reader is referred to Section 6.10.10 of the
AASHTO LRFD BDS.
Between points of zero and maximum positive moment, the required Strength
limit state design shear, P1, is based on PP and FP, and n gives the number of studs
required between those points. Between points of maximum negative and maximum
positive moment, the required Strength limit state design shear, P2, is based on PT
and FT, and n gives the number of studs required between those points.
Qr = fscQn = 0.85Qn
(6.77)
Qn = 0.5 Asc fc¢Ec £ Asc Fu
(6.78)
n=
P
Qr
(6.79)
P1 = PP2 + FP2
(6.80)
P2 = PT2 + FT2
(6.81)
0.85 fc¢bs t s
ì
PP = Min í
î Dt w Fyw + Fyt b ft t ft + Fycb fct fc
(6.82)
148
LRFD Bridge Design
PT = PP + Pn
(6.83)
0.45 fc¢bs t s
ì
Pn = Min í
î Dt w Fyw + Fyt b ft t ft + Fycb fct fc
(6.84)
6.9 PLASTIC MOMENT COMPUTATIONS
Plastic moment computations are based on yielding of the entire cross section, not
simply the most remote fibers of the cross section. Concrete stress is set to be equal
to 0.85f’c and all steel elements, including any reinforcement included in the calculation, are stressed to their respective yield stress, f y. Under the full yielding crosssectional assumption, the location within the cross-section depth at which the forces
above exactly equal the forces below is the plastic neutral axis (PNA).
For non-composite sections, the calculations are based solely on the girder properties and are rather straightforward.
For composite sections in negative flexure, given that the plastic moment is a
Strength or Extreme Event limit state, the concrete deck is typically not included,
but the deck reinforcement may be included in cases where adequate shear studs are
provided in the negative moment region.
For composite sections in positive flexure, it is not uncommon to discount any
reinforcement in the concrete deck since such reinforcement is typically reduced
from that in negative moment regions so that a smaller influence on the deck reinforcement occurs. Nonetheless, deck reinforcement may be included, provided the
construction documents clearly outline the required reinforcing details. Certainly,
the concrete deck is included in plastic moment calculations for girders designed to
act composite with the deck through adequate shear stud provision.
6.10 SOLVED PROBLEMS
Problem 6.1
Using the NSBA LRFD-Simon software, create a continuous beam model
of the steel superstructure for an interior girder of the Project Bridge. Use a
welded plate I-girder with constant properties over the entire bridge length.
In practice, design optimization would include flange thickness and/or
width transitions to avoid material waste. For this problem use the following properties and parameters:
• bf = 16 inches, tf = 1.25 inches, top and bottom flanges
• D = 46 inches, t w = 0.50 inches, web
• Fy = 50 ksi, Fu = 70 ksi, flanges and web
• f’c = 4 ksi, n = 8, 3n = 24, composite deck
• As = 9.2 in2, deck reinforcement in negative moment regions
• Top of the web to the bottom of the deck = 3.25 inches
• Assume metal deck forms add a weight equal to 1 inch of concrete
• Use weathering steel for all plates
• Pour #1 is from the ends of the bridge to the field splice
• Pour #2 is from the field splice to the pier
Steel Welded Plate I-Girders
149
• Parapet weight = 0.400 klf per parapet
• Girder spacing S = 9 ft 3 inches
• Try for an unstiffened web design
Problem 6.2
Using the results reported by Simon for Problem 6.1, verify the section
properties for the girder, the girder with rebar, the short-term composite
properties, and the long-term composite properties.
Problem 6.3
Using the reported moments from Simon for Problem 6.1, check the stresses
reported for the Strength I limit state at the points of maximum positive
moment and maximum negative moment.
Problem 6.4
Using the reported shear results from Problem 6.1, check the reported
shear resistances at the end of the bridge and at the pier.
Problem 6.5
Using results from the Simon model in Problem 6.1, verify the Fatigue I
limit state reported ratios for each of the following at the maximum positive
moment section:
a) base metal, bottom flange, Category B
b) web-to-flange weld, bottom flange, Category B
c) stiffener-to-flange weld, bottom flange, Category C’
Problem 6.6
Use three 0.75-inch diameter studs per row on the top 16-inch × 1.25-inch
flange for the Project Bridge steel girder. Using fatigue shear range
and section property results from the Simon model from Problem 6.1,
determine:
• the shear stud pitch requirements for the Fatigue I limit state
• the number of shear studs between points of zero and maximum positive moment for the Strength limit state
• the number of shear studs between the points of maximum positive and
maximum negative moment for the Strength limit state.
Problem 6.7
For the steel girder of the Project Bridge, use Simon results from Problem
6.1 to verify the moment resistance at the section 36.59 feet from the abutment end. The following parameters are given for the plastic condition;
other necessary parameters are to be found in the Simon results.
• MP = 9,025 ft-kips
• yP = 48.49 inches, bottom of girder to plastic neutral axis (PNA)
Problem 6.8
A welded steel plate I-girder consists of 75-inch × 0.50-inch web with
20-inch × 1-inch flanges, top and bottom. All I-girder plates are Grade 70W
weathering steel. Assume that the required design shear is equal to the
design shear resistance and the design two-sided transverse stiffeners using
Grade 50W steel. Check two-sided bearing stiffeners, 9.5 inches × 0.875
inches, Grade 50W, with 1-inch clips, for a Strength limit state reaction at
an interior Pier of 900 kips.
150
PROBLEM 6.1
LRFD Bridge Design
TEH
1/5
151
Steel Welded Plate I-Girders
PROBLEM 6.1
TEH
2/5
152
PROBLEM 6.1
LRFD Bridge Design
TEH
3/5
153
Steel Welded Plate I-Girders
PROBLEM 6.1
TEH
4/5
154
PROBLEM 6.1
LRFD Bridge Design
TEH
5/5
155
Steel Welded Plate I-Girders
PROBLEM 6.2
TEH
1/4
156
PROBLEM 6.2
LRFD Bridge Design
TEH
2/4
157
Steel Welded Plate I-Girders
PROBLEM 6.2
TEH
3/4
158
PROBLEM 6.2
LRFD Bridge Design
TEH
4/4
159
Steel Welded Plate I-Girders
PROBLEM 6.3
TEH
1/2
160
PROBLEM 6.3
LRFD Bridge Design
TEH
2/2
161
Steel Welded Plate I-Girders
PROBLEM 6.4
TEH
1/2
162
PROBLEM 6.4
LRFD Bridge Design
TEH
2/2
163
Steel Welded Plate I-Girders
PROBLEM 6.5
TEH
1/2
164
PROBLEM 6.5
LRFD Bridge Design
TEH
2/2
165
Steel Welded Plate I-Girders
PROBLEM 6.6
TEH
1/5
166
PROBLEM 6.6
LRFD Bridge Design
TEH
2/5
167
Steel Welded Plate I-Girders
PROBLEM 6.6
TEH
3/5
168
PROBLEM 6.6
LRFD Bridge Design
TEH
4/5
169
Steel Welded Plate I-Girders
PROBLEM 6.6
TEH
5/5
170
PROBLEM 6.7
LRFD Bridge Design
TEH
1/1
171
Steel Welded Plate I-Girders
PROBLEM 6.8
TEH
1/3
172
PROBLEM 6.8
LRFD Bridge Design
TEH
2/3
173
Steel Welded Plate I-Girders
PROBLEM 6.8
TEH
3/3
174
LRFD Bridge Design
6.11 EXERCISES
E6.1
A welded plate girder made from Grade 50W weathering steel has a
52-inch × 1/2-inch web. Compute the design shear resistance, ϕVn, for each
of the following conditions:
a) an unstiffened end panel
b) a stiffened end panel with stiffener spacing, do = 1.5D
c) a stiffened interior panel with stiffener spacing, do = 1.5D
E6.2
Using the reported moments from the Simon model in Problem 6.1, verify
the reported stresses in the flanges over the Pier at the Strength I limit state
for the Project Bridge.
E6.3
For the Project Bridge, a field splice using 16-inch × 3/4-inch outer plate
at each flange and two 6.5-inch × 1-inch inner plates at each flange is proposed. The proposed web splice plates are 42 inch × 3/8 inch. Use Grade
50W splice plates and assume four bolts (A325, threads included, and
Class B surface) across the width of the flange. The girder consists of
16-inch × 1.25-inch flanges and a 46-inch × 0.50-inch web, all made from
Grade 50W plate. Assume that the web is unstiffened at the splice point.
• Check the flange splice plate areas for total area requirements and for
equal distribution of design force requirements.
• Check the web splice plate areas for total area requirements.
• Determine the flange splice design force, Pfy.
• Determine the Strength I limit state maximum and minimum moments
at the splice point, using Simon results from Problem 6.1.
• Determine the moment resistance provided by the flanges.
• Determine the web design force, Rweb.
• Determine the number of flange bolts and the number of web bolts
based strictly on bolt shear.
• Enter the data onto the NSBA-Splice spreadsheet to perform a complete
check of the proposed design.
E6.4
A welded plate I-girder is used as the design basis for a three-span
bridge. Span lengths are 234 ft, 300 ft, and 234 ft. At a particular point
on the span, the girder cross section consists of 103-inch × 0.75-inch web,
18-inch × 1-inch top flange, 20-inch × 1.0625-inch bottom flange, and 8-inch
concrete deck. The distance from the top of the web to the bottom of the
deck is 5.5 inches. The unfactored moments, per girder, are summarized as
follows:
• MDC1 = 4,823 ft-kips
• MDC2 = 448 ft-kips
• MDW = 1,137 ft-kips
• MLL+IM = 7,945 ft-kips, −2,720 ft-kips
• MFatigue = 2,349 ft-kips, −713 ft-kips (both values include impact)
175
Steel Welded Plate I-Girders
TABLE E6.4
Exercise E6.4
Property
Girder
Short-term Composite
Long-term Composite
I, in4
174,249
431,434
312,546
yb, inches
Stop, in3
51.11
3,229
86.55
23,309
70.20
8,965
Sbott, in3
3,410
4,985
4,452
The elastic section properties of the girder at various stages are given in
Table E6.4. Yield moment, plastic moment, and distance from the girder
bottom to the PNA are:
• My = 17,713 ft-kips
• MP = 30,599 ft-kips
• yP = 104.2 inches
Determine each of the following:
• the stresses at the bottom and the top of the girder at the Strength limit
state.
• the fatigue stress range and acceptable category for a connection stiffener to bottom flange weld for the Fatigue I (infinite life) limit state.
• the required moment resistance and design moment resistance for the
compact section in positive flexure at the Strength limit state.
E6.5
A three-span, steel I-girder bridge consists of the cross section shown in
Figure E6.5 with the following parameters for an interior girder:
• beff = 126 inches
• ts = 8 inches
• x = 5.5 inches, from the top of the web to the bottom of the deck
FIGURE E6.5
Exercise E6.5.
176
LRFD Bridge Design
TABLE E6.5
Exercise E6.5
Plate
Left of Splice Centerline
Right of Splice Centerline
Top flange (inches)
22 × 1.5625
19 × 0.8750
Web (inches)
109 × 0.7500
109 × 0.750
Bottom flange (inches)
24 × 2.2500
19 × 0.9375
Span lengths are 234 ft, 300 ft, and 234 ft. The cross section to the left and
the right of the centerline of a field splice 75 feet into span 2 are tabulated
in Table E6.5.
The concrete for the deck has a 28-day minimum strength of 5 ksi. All
plates are Grade 50W steel. Unfactored moments at the splice location are
as follows:
• MDC1 = −741 ft-kips, non-composite
• MDC2 = −7 ft-kips, long-term composite
• MDW = −13 ft-kips, long-term composite
• MLL + IM = 5,252 ft-kips (maximum) and −4,392 ft-kips (minimum),
short-term composite
• MDeckPour = −3,810 ft-kips
Include effects from any required filler plates, and (a) check the proposed
splice plates for required area and equal load distribution and (b) set the
number of bolts for shear. Enter the data onto the NSBA Splice spreadsheet and perform a full check on the proposed splice design. The proposed
splice design is summarized below.
• Outer splice plates for flange = 19 × 9/16, top and bottom
• Inner splice plates for flange = 8.125 × 5/8, top and bottom
• Web splice plates = 102.75 × 3/8, stiffener spacing do = 12 ft
• 7/8-inch A325-N bolts, 4 rows for flanges and 2 rows each side of the web
E6.6
The framing plan for the center 300-ft span of a three-span bridge is shown
in Figure E6.6a. A typical intermediate cross-frame between any two adjacent girders is shown in Figure E6.6b. The moments during construction are
estimated to be 3,729 ft-kips from component dead load and 4,100 ft-kips
from estimated construction live loads prior to deck curing. The chord and
diagonal members of the cross-frames each have an area equal to 5.00 in2.
Determine the web distortional stiffness, βsec, the girder system stiffness,
βg, and the cross-frame stiffness, βb. Compute the actual total system stiffness and the required system stiffness, βT. Determine the K-frame chord
and diagonal design forces. Additional data are summarized below.
• Web plate = 109 inches × 0.75 inches
• Flange plate = 30 inches × 1.25 inches, top and bottom
• Stiffeners = 8 inches × 0.75 inches, two sided
Steel Welded Plate I-Girders
FIGURE E6.6
177
Exercise E6.6.
This problem is somewhat academic, in that the flange width of 30 inches is
likely larger than would be used in practice by most engineers, but serves to
illustrate the calculations. A more probable solution might be to use lateral
bracing in two or three bays adjacent to each support (abutments and piers)
to reduce the effective span for torsional brace (cross-frame) design.
E6.7
A welded steel plate girder made from steel with Fy = 50 ksi is used for an
I-girder bridge. Girder spacing, S, = 12 ft, deck thickness, ts = 8 inches, and
concrete strength, f’c = 4 ksi. The top flange is 16 inches × 0.8125 inches and
the bottom flange is 16 inches × 1.125 inches. The web is 96 inches × 0.6875
inches. The distance from the top of the web to the bottom of the deck is 3
inches. For positive flexure (compression in the top of the deck), determine
the plastic moment, Mp, and the yield moment, My. Properties (section moduli at the top of the girder and the bottom of the girder) at various stages are
provided in Table E6.7. Unfactored moments are:
• MDC1 = 3,260 ft-kips (carried by the girder alone)
• MDC2 = 661 ft-kips (carried on the long-term composite section)
• MDW = 882 ft-kips (carried on the long-term composite section)
E6.8
A welded steel plate girder made from steel with Fy = 50 ksi includes 12 in2
of reinforcing in the deck over the interior supports in the negative moment
region. The top flange is 20 inches × 2.75 inches and the bottom flange is
178
LRFD Bridge Design
TABLE E6.7
Exercise E6.7
Property
Girder
Composite (n)
Composite (3n)
I, in4
122,983
315,538
229,493
3
Stop, in
2,396
18,599
7,106
Sbott, in3
2,639
3,897
3,496
22 inches × 2.75 inches. The web is 96 inches × 0.6875 inches. Deck reinforcing is located 4.75 inches above the top of the girder and f y = 60 ksi
for the bars. Compute the plastic and yield moments in negative flexure.
Unfactored moments are as follows:
• MDC1 = 9,176 ft-kips (carried by the girder alone)
• MDC2 = 1,672 ft-kips (carried on the long-term composite section)
• MDW = 2,229 ft-kips (carried on the long-term composite section)
For the girder alone, I = 331,931 in4 and the center of gravity is located
49.25 inches from the bottom of the girder.
E6.9
Suppose the Project Bridge is located at a site characterized by surface
category D conditions for wind load calculations, with V = 115 mph for the
Strength III limit state from maps, and height, Z = 35 feet. Check the flexural resistance at the Strength I, Strength III, and Strength V limit states for
the maximum negative moment section over the pier. Cross section details
and moments from a Simon model are given below. All steel is Grade 50W
weathering steel. The LTB modification factor, Cb, may be taken to be equal
to 1.36 for all calculations.
• Web plate, 36 inch × 0.65 inches thick
• Flange plates, 14 inch × 1.50 inches thick
• Cross-frame spacing adjacent to the pier = 15 feet (Figure 1.2)
• Reinforcing bars in deck, As = 10 in2, 7.5 inches above the top of the web
• MDC1 = −1,273 ft-kips, non-composite
• MDC2 = −202 ft-kips, composite
• MDW = −283 ft-kips, composite
• MLL + IM = −1,366 ft-kips, composite
7
Precast Prestressed
Concrete Girders
Precast, prestressed concrete girders are efficient, economical options for bridges
with spans of less than about 150 feet in many locations. Some states, Florida and
Washington for example, do have deep precast concrete girders capable of spanning
much greater distances. Spliced girder designs have also been used to extend span
capabilities for prestressed girders. However, when spans are longer than about 150
feet, steel girders become the preferred choice in areas where the standard AASHTO
I-beam and Bulb-T beam cross sections are the only readily available concrete
I-girder sections.
The basic idea underlying prestressed girder design is to fully or partially offset the effects of gravity loading with opposing moments and axial loads. At midspan, gravity loads impart tension into the bottom flange of an I-girder. Prestressing
applied below the centroid of the girder produces compression in the bottom flange
through both a compressive axial force and a negative moment due to the eccentricity of the axial force.
At girder ends made continuous and composite with the concrete deck over interior supports, negative moment reinforcement in the deck, coupled with girder strand
extensions into a cast-in-place concrete diaphragm, are relied upon to provide the
continuity.
Two of the most commonly used strands are 0.50-inch diameter (area = 0.153 in2)
and 0.60-inch diameter (area = 0.217 in2) low-relaxation seven-wire strands with tensile strength, fpu = 270 ksi. For these strands, E = 28,500 ksi and f py = 0.90fpu.
Chapter 1 of this text includes a presentation of standard, prestressed girder
shapes and properties.
7.1 STRESS ANALYSIS
Figure 7.1 depicts the general stress states from various loading sources on a noncomposite precast, prestressed concrete (PPC) girder (prior to it becoming composite with the concrete deck).
AASHTO requirements for PPC girders include Service limit state stress limits
in addition to Strength limit state flexural resistance.
Some owners also require that the net deflection due to all load sources be upward,
not downward. The requirement is sometimes specified to be satisfied using deflection multipliers equal to 1.0 on all sources, rather than the default values in many
software packages.
Stress limits are specified in the AASHTO LRFD BDS both at strand detensioning and for the in-place girder, subject to full design loads. Stress limits at
DOI: 10.1201/9781003265467-7
179
180
LRFD Bridge Design
FIGURE 7.1 Precast Girder Stresses.
de-tensioning are based on non-composite girder properties along with the specified
minimum girder release concrete strength, f’ci. Final condition stress limits are based
on composite section properties and the final required minimum concrete strength
of the girder, f’c.
For temporary stresses prior to losses, concrete compression is limited to 0.65 f’ci.
For temporary stresses prior to losses, for normal-weight concrete in areas with
bonded reinforcement (bars or strands) sufficient to resist concrete tensile force, concrete tension is limited to 0.24 fci¢ .
Final stresses for normal-weight concrete, after all losses have occurred, are limited to the values shown in Table 7.1. The reduction factor, ϕw, is applicable to box
girders and is equal to 1.0 whenever the flange and web slenderness values are both
less than 15. Refer to Section 5.9.2.3.2 of the AASHTO LRFD BDS for other conditions of web and flange slenderness.
For stress calculations at release, one recommended method suggests the use of
transformed properties along with the initial pull, with elastic shortening loss effects
ignored (Swarz et al., 2012).
7.2 FLEXURAL RESISTANCE
For positive flexure (compression in the bridge deck) at the Strength limit state,
Equation 7.1 provides the theoretical nominal resistance for the simplified case in
which any mild reinforcement in the tension region of the beam and any compression reinforcement in the slab are both ignored. If the stress block depth, given by
Equation 7.3, is less than the deck thickness, then (b−bw) = 0 and the entire second
term on the right-hand side of Equation 7.1 is zero. Aps is the total area of strands, f ps
is the strand stress at nominal resistance, b is the effective width of the deck, and dp is
the distance from the top of the deck to the centroid of the strands. The deck concrete
strength, rather than the girder concrete strength, is to be used in the equations,. The
181
Precast Prestressed Concrete Girders
TABLE 7.1
Final Stress Limits in Prestressed Concrete Bridge I-Girders
Limit State and Loading Conditions
Service I limit state compression
Effective prestress plus permanent load
Service I limit state compression
Effective prestress, permanent loads, transient loads
Service III limit state tension w/ bonded reinforcement
Moderate corrosion conditions
Service III limit state tension w/ bonded reinforcement
Severe corrosion conditions
Limiting Stress
0.45f’c
0.60ϕwf’c
0.19 fc¢ £ 0.600 ksi
0.0948 fc¢ £ 0.300 ksi
coefficient, k, in the equations is 0.28 for Low-Lax strands. The resistance factor, ϕ,
for tension-controlled, prestressed concrete beams is 1.0.
aö
æ
M n = Aps f ps ç d p - ÷ + a1 fc¢ ( b - bw ) h f
2ø
è
c=
æ a - hf ö
ç 2 ÷
ø
è
Aps f pu
a1 fc¢b1b + kAps f pu / d p
(7.1)
(7.2)
a = b1c
(7.3)
0.85, fc¢ £ 10 ksi
ì
ï
a1 = í
0.75, fc¢ ³ 15 ksi
ïinterpolate for intermediate vaalues
î
(7.4)
0.85, fc¢ £ 4 ksi
ì
ï
b1 = í
0.65, fc¢ ³ 8 ksi
ïinterpolate for intermediate valu
ues
î
(7.5)
æ
c ö
f ps = f pu ç 1 - k ´ ÷
d
p ø
è
(7.6)
æ
f ö
k = 2 ç 1.04 - py ÷
f pu ø
è
(7.7)
ì0.90, strands
f py ï
= í0.85, plain bars
f pu ï
î0.80, deformed bars
(7.8)
182
LRFD Bridge Design
For negative flexure over interior piers, a strand stress equal to zero at the girder ends
may be used, and negative moment deck reinforcement for the composite beam is
designed, using reinforced concrete principles.
7.3 SHEAR RESISTANCE
The summary of shear provisions for prestressed girders presented here is applicable
to so-called “B-regions” without significant torsion. These are regions where the
assumption that plane sections remain plane is judged to be valid. For deep beams
and other disturbed regions (“D-regions”) the reader is referred to Section 5.8 of
the AASHTO LRFD BDS. Typical, prestressed concrete bridge girders are usually
designed as B-regions, provided all requirements in Section 5.7 of the AASHTO
LRFD BDS are satisfied.
The shear depth, dv, is taken to be equal to the distance between the cross-section
resultant tensile and compressive forces but need not to be taken to be less than either
0.72h or 0.90de; de is the distance from the extreme compression fiber to the resultant
cross-section tensile force and h is the total member depth. The shear width, bv, is
the minimum web width.
Nominal shear resistance, Vn, consists of contributions from the concrete, from
the steel shear reinforcement, and from the vertical component of prestress force at
a given section. Equation 7.9 provides the nominal resistance. The units on f’c must
be ksi in the following equations. For normal-weight concrete with stirrups inclined
at 90 degrees to the longitudinal axis of the beam, Equations 7.10 and 7.11 provide
the concrete and steel contributions, respectively. Vp is the vertical component of
prestress force at the section under consideration.
Vn = Vc + Vs + Vp £ 0.25 fc¢bv dv
(7.9)
Vc = 0.0316 b fc¢bv dv
(7.10)
Vs =
Av f y dv
× cot q
s
(7.11)
The minimum area of shear reinforcement is required whenever Vu > 0.5ϕ(Vc+ Vp)
and is given by Equation 7.12 for normal-weight concrete. The resistance factor for
shear is 0.85.
Av ³ 0.0316 fc¢ ×
bv s
fy
(7.12)
Shear reinforcement spacing limits depend on the shear stress on the concrete,
vu, which is calculated using Equation 7.13. The spacing limits are given in
Equation 7.14.
vu =
Vu - fVp
f bv dv
(7.13)
183
Precast Prestressed Concrete Girders
ì0.4dv £ 12 inches, if vu ³ 0.125 fc¢
smax = í
î0.8dv £ 24 inches, if vu < 0.125 fc¢
(7.14)
Calculation of the coefficients β and θ is required to accurately assess shear resistance. These factors depend on the net tensile strain in the centroid of the tension
reinforcement, εs. The discussion here is limited to sections with at least the minimum amount of transverse shear reinforcement. For such cases, the strain and the
coefficients are given by Equations 7.15 through 7.17. The strain may also be determined by a cross-sectional analysis. For other cases, the reader is referred to Section
5.7.3.4.2 of the AASHTO LRFD BDS.
æ Mu
ö
+ 0.5N u + Vu - Vp - Aps f po ÷
çç
÷
dv
ø
es = è
Es As + E p Aps
b=
4.8
1 + 750e s
q = 29 + 3500e s
(7.15)
(7.16)
(7.17)
For typical values of prestressing parameters, f po may be taken to be equal to zero
at the point of bonding between strands and surrounding concrete up to 0.7fpu at the
transfer length. Refer to Section 5.7.3.4.2 of the AASHTO LRFD BDS for additional
discussion on fpo.
For calculated strains, εs, of less than zero, εs may be taken to be equal to zero.
The upper limit on εs to be used for calculation of shear resistance coefficients is εs
≤ 0.006. This effectively places a lower limit on β equal to 0.873 and an upper limit
on θ equal to 50 degrees. Other limitations on the parameters are summarized below.
• Nu, the factored axial force on the cross section, is positive if tension is
applied, negative if compression is applied.
• Mu, the factored moment at the section, is not to be less than (Vu −Vp) dv.
• For sections closer than dv to the face of support, εs, at a distance dv from the
face of support, may be used to determine β and θ, unless a concentrated
load is located within dv from the support, in which case εs should be determined at the face of support.
For additional shear-related longitudinal reinforcement requirements, refer to Section
5.7.3.5 of the AASHTO LRFD BDS.
Section 5.7.4 of the AASHTO LRFD BDS covers interface shear reinforcement
requirements. Regarding prestressed bridge girders, the provisions are applicable to
horizontal shear transfer between the precast, prestressed girder and the cast-in-place
concrete deck. For such members, interface shear transfer is typically accomplished
by extending a portion of the girder shear stirrups into the deck with terminating
hooks.
184
LRFD Bridge Design
The minimum required area of interface shear reinforcement is given by Equation
7.18. The value for f y in Equations 7.18 and 7.19 is not to exceed 60 ksi, regardless of
the reinforcement grade used for shear steel. The design interface shear resistance
is given by Equation 7.19. Pc is the permanent net compressive force normal to the
shear plane.
Avf =
0.05 Acv
fy
(7.18)
fVni = f ( cAcv + m Avf f y + m Pc ) £ f K1 fc¢Acv £ f K 2 Acv
(7.19)
Table 7.2 summarizes μ, c, K1, and K2 for various conditions using normal-weight
concretes. For conditions with lightweight concretes, refer to Section 5.7.4.4 of the
AASHTO LRFD BDS.
The maximum spacing of interface shear transfer reinforcement is limited to 48
inches or the girder depth, whichever is smaller.
7.4 CONTINUITY DETAILS
Prestressed girders are typically constructed to span between successive supports
(abutments or piers) as simply supported beams for self-weight, wet deck concrete
weight, intermediate diaphragms, and construction loads prior to deck concrete curing. Prestressed concrete girders running continuously over interior supports require
longitudinal deck reinforcement, cast-in-place support diaphragms, and the extension of girder strands into the diaphragm to enable the composite girder to resist
negative moments. Refer to the LRFD BDS, Section 5.12.3.3, for additional details
on continuity not fully presented here.
TABLE 7.2
Interface Shear Parameters
μ
c
K1
K2
Cast-in-place concrete slab on clean concrete girder
surfaces, free of laitance, with surface intentionally
roughened to an amplitude of 0.25 inches
1.0
0.28 ksi
0.30
1.8 ksi
Normal-weight concrete placed monolithically
Normal-weight concrete placed against a clean concrete
surface, free of laitance, with surface intentionally
roughened to an amplitude of 0.25 inches
Concrete placed against a clean concrete surface, free of
laitance, but not intentionally roughened
1.4
1.0
0.40 ksi
0.24 ksi
0.25
0.25
1.5 ksi
1.5 ksi
0.6
0.075 ksi
0.20
0.80 ksi
0.7
0.025 ksi
0.20
0.80 ksi
Conditions
Concrete anchored to as-rolled structural steel by headed
studs or by reinforcing bars, where all steel in contact
with concrete is clean and free of paint
Precast Prestressed Concrete Girders
185
Restraint moments from time-dependent deformations develop in precast girders
made continuous. These restraint moments can be ignored in design as long as the
girder age, at the time continuity is achieved, is 90 days or more. For other cases,
refer to the LRFD BDS, Section 5.12.3.3.2 and Commentary, for restraint moment
calculations to be accounted for in the design. Restraint moments may be positive or
negative. Positive restraint moments at girder ends may reduce the effectiveness of
the continuity diaphragm by producing excessive cracking at the compression block.
Hence, if restraint moments are neglected in the girder design, a plans-note stating
that the girder age must be no less than 90 days at the time continuity is established
will be required.
Even when girder age is specified on the plans to be no less than 90 days at continuity, the LRFD BDS requires that a positive moment connection with a design
resistance, ϕMn, no less than 1.2 times the cracking moment, Mcr, be provided. Such
positive moment connections may consist of at least three options.
1. Non-prestressed reinforcement embedded in the girder and developed into
the continuity diaphragm
2. Extension of prestressing strands beyond the end of the girder and anchored
into the continuity diaphragm.
3. Testing to validate other methods an owner may propose.
Strands de-bonded at girder ends to control tensile stresses in the girder are not to be
used as elements contributing to the positive moment connection.
For normal-weight concrete with specified strength of no more than 15 ksi, the
cracking moment, Mcr = 0.24(f’c)0.5(Ig)/yt, is to be computed using the composite
girder/deck geometric properties, but with a concrete strength corresponding to that
of the continuity diaphragm, which is generally cast simultaneously with the deck
over the support.
When projecting strands are used for the positive moment connection, the LRFD
BDS, in Sections 5.12.3.3.9c and 5.12.3.3.9d, requires that several criteria be satisfied.
• Extended strands must be anchored into the continuity diaphragm with
either 90-degree hooks or a strand development length.
• Extended strands must extend no less than 8 inches from the end of the
girder prior to bending.
• Extended strands shall form a generally symmetrical pattern about the centerline of the girder.
• Strands from opposing girders must be detailed so as to preclude conflicts
in the reinforcement patterns.
For stress in the extended strands, Equations 7.20 (for the Service limit state)
and 7.21 (for the Strength limit state) from the LRFD BDS Commentary can be
used. Equations 7.20 and 7.21 were developed based on 0.5-inch-diameter strands.
Although not in the LRFD BDS, equations for 0.6-inch-diameter strands used in the
original report (Miller, et al., 2004) for continuity calculations are presented here as
186
LRFD Bridge Design
Equations 7.22 and 7.23, based on development lengths proportional to strand diameters. The total length of the extended strand, ldsh, includes both the straight and bent
portions of the strand. The resulting calculated stresses are in ksi.
f psl
f pul
f psl =
( ldsh - 8 ) ,
0.50-inch strand
(7.20)
f pul =
( ldsh - 8 ) , 0.50-inch strand
(7.21)
0.228
0.163
æ æ 0.50 ö ö
ç ldsh ç
÷ -8÷
è 0.60 ø ø
=è
, 0.60-inch strand
0.228
æ æ 0.50 ö ö
ç ldsh ç
÷ -8÷
è 0.60 ø ø
è
=
, 0.60-inch strand
0.163
(7.22)
(7.23)
With girder age no less than 90 days at the time continuity is established, the positive
connection design moments may be taken to be equal to Mcr at the Service limit state
and 1.2Mcr at the Strength limit state. Although not explicitly stated in the AASHTO
LRFD BDS, these design moments were used in the research upon which the LRFD
BDS provisions are based.
The original research (Miller et al., 2004) upon which the LRFD BDS provisions
are based recommends a total strand embedment, ldsh, such as to produce a Service
limit state stress, fpsl, no greater than 150 psi. This results in a 42-inch embedment for
0.5-inch-diameter strands and a 51-inch embedment for 0.60-inch-diameter strands.
Of course, when analysis indicates that a shorter embedment satisfies all requirements, a shorter embedment may be used. Little, if any, benefit is gained by using
larger than necessary embedment, it seems.
7.5 MILD TENSILE REINFORCEMENT IN GIRDERS
Prestressed concrete I-girders, whether continuous or not for live loads, typically
require non-prestressed reinforcement in the top flange of the girder to resist tensile
stresses at strand release.
The required area of steel in the girder top flange may be calculated using the
method proposed in the LRFD BDS Commentary to Section 5.9.2.3.1b. The stresses
required for the calculations are those in the top, fcitop, and bottom, fcibot, of the girder
at strand release prior to losses in strand stress. With reference to Figure 7.1, these
stresses at the girder ends can be approximated to by Equations 7.24 and 7.25, with
the required area of steel, As, given by equation 7.26. The parameter, btop, is the top
flange width and x is the distance from the top of the girder to the location of zero
187
Precast Prestressed Concrete Girders
stress in the girder. The total height of the non-composite girder is h. Cross-sectional
properties of the girder include the area, Ag, the moment of inertia, Ig, and the distances from the neutral axis to the top, ct, and bottom, cb, of the girder. Pi is the total,
initial, prestress force and e is the distance from the centroid of the strands to the
girder neutral axis.
fcitop =
Pi ( Pi ´ e ) ct
Ag
Ig
(7.24)
fcibot =
Pi ( Pi ´ e ) cb
+
Ag
Ig
(7.25)
As =
fcitopbtop x
2 fs
(7.26)
æ
ö
fcitop
x =ç
÷h
f
f
è cibot citop ø
(7.27)
fs = 0.50 f y £ 30 ksi
(7.28)
The sign convention used here is negative for tensile stress. Should the calculated
stress in the top of the girder, fcitop, be positive (compression), then the theoretical
area of mild reinforcement required is zero. The sign on the stresses calculated
should be retained to properly evaluate the equations.
7.6 NEGATIVE MOMENT REINFORCEMENT FOR
GIRDERS MADE CONTINUOUS
For precast, prestressed girders made continuous for composite dead load and live
load, deck reinforcement is required to carry tension in the regions for which negative moment exists in the continuous beam. Such reinforcement is typically epoxy
coated and may consist of longitudinal bars in the deck in two mats. Careful attention to bar size is necessary, given the congestion possible in relatively thin bridge
decks with two mats of steel running in each direction.
Some states permit the use of partial depth, precast panels for bridge decks,
at the contractor’s option. In such cases, the contractor is sometimes permitted to
omit the bottom mat of longitudinal reinforcing in the deck. Whenever the bottom mat is counted on as part of the negative moment reinforcing, however, such
omission must not be permitted. The engineer must either use only the top mat
of longitudinal bars to carry the negative moment tension, or disallow the use of
precast panels.
Negative moment reinforcement in the deck is subject to Strength, Service, and
Fatigue limit state requirements in the AASHTO LRFD BDS.
188
LRFD Bridge Design
Strength limit state flexural resistance may be computed using Equation 7.29,
based on the simplifying assumptions that (a) the effects of any mild, compression reinforcement in the bottom of the beam may be ignored, and (b) the tension
steel has yielded (this must always be verified, or adjustment made to the calculated
resistance).
The flange thickness, hf, is that for the bottom of the prestressed beam. The flange
width, b, is for the bottom of the prestressed beam, and the web width, bw, is for the
prestressed beam. The depth from the extreme compression fiber (at the bottom of
the beam) to the centroid of the negative moment reinforcing in the deck is ds. If the
calculated stress block depth, a, is less than hf, then the entire second term on the
right-hand side of Equation 7.29 is zero, as is the second term in the right-hand side
numerator of Equation 7.30, since (b−bw) is zero in such a case.
Given that the maximum negative moment occurs at the centerline of the interior
supports (piers or bents), one strategy is to incorporate in the equations the compressive strength of the cast-in-place diaphragm (which is typically the same concrete as
used for the deck and is cast simultaneously with the deck) rather than that for the
beam. This is recommended practice, even though the bottom of the beam is under
compression, since the compressive stress block also exists in the cast-in-place diaphragm (at least at the end of the girder).
é
aö
æ a - hf
æ
f M n = f ê As f y ç ds - ÷ + a1 fc¢ ( b - bw ) h f ç
2
è
ø
è 2
ë
c=
As f y - a1 fc¢ ( b - bw ) h f
a1 fc¢b1bw
öù
÷ú
øû
(7.29)
(7.30)
a = b1c
(7.31)
0.85, fc¢ £ 10 ksi
ì
ï
a1 = í
0.75, fc¢ ³ 15 ksi
ïinterpolate for intermediate vaalues
î
(7.32)
0.85, fc¢ £ 4 ksi
ì
ï
b1 = í
0.65, fc¢ ³ 8 ksi
ïinterpolate for intermediate valu
ues
î
(7.33)
Negative moment, deck reinforcement design for precast beams made continuous
often results in a tension-controlled section, with a corresponding resistance factor
(ϕ) equal to 0.90. This must be verified by checking the strain in the tension reinforcement, and if the section is not tension-controlled, then the resistance factor must
be reduced. The resistance factor for compression-controlled sections is 0.75. For
sections neither tension controlled nor compression controlled, linear interpolation
is used to determine the appropriate ϕ factor.
189
Precast Prestressed Concrete Girders
Tension-controlled reinforced concrete sections are defined as sections with a net
tensile strain in the extreme layer of tensile reinforcement, εt, greater than or equal to
the tension-controlled strain limit, εtl, when the concrete strain reaches a value of 0.003.
Compression-controlled reinforced concrete sections are defined as sections with a
net tensile strain in the extreme layer of tensile reinforcement, εt, less than or equal to the
compression-controlled strain limit, εcl, when the concrete strain reaches a value of 0.003.
The tension-controlled strain limit, εtl, is determined as follows:
• εtl = 0.005 for reinforcement with f y ≤ 75 ksi
• εtl = 0.008 for reinforcement with f y = 100 ksi
• εtl is determined by linear interpolation for reinforcement with 75 < f y <
100 ksi
The compression-controlled strain limit, εcl, is determined as follows:
• εcl = f y /Es, but not > 0.002, for reinforcement with f y ≤ 60 ksi
• εcl = 0.004 for reinforcement with f y = 100 ksi
• εcl is determined by linear interpolation for reinforcement with 60 < f y <
100 ksi
For the Service limit state, negative moment deck reinforcing is subject to the crack
control requirements of the AASHTO LRFD BDS, Section 5.6.7. The lateral spacing
(s) of longitudinal bars must not exceed the value given by Equation 7.34. The exposure factor, γe, is 1.0 for Class 1 exposure conditions and 0.75 for Class 2 exposure
conditions. Decks are typically assigned Class 2 exposure due to their susceptibility
to corrosive conditions. Some owners choose to adopt an intermediate value for γe.
The parameter, βs, is the ratio of the distance from the cracked neutral axis to the
extreme tension face to the distance from the cracked neutral axis to the centroid of
reinforcement closest to that face. The actual definition involves strains, but, under the
assumption that strains are proportional to distance from the neutral axis, the definition provided is equivalent. An estimate of the value may be calculated from Equation
7.35, or a more precise value may be obtained from strain compatibility analysis.
The stress, fss, in the tension steel at the Service limit state, is to be computed
based on cracked section properties, and is not to exceed 0.6f y. The overall height of
the member is h, and the distance from the extreme tension face to the centroid of the
reinforcement closest to that face is dc.
s£
700g e
- 2 dc
b s fss
bs = 1 +
dc
0.7 ( h - dc )
(7.34)
(7.35)
Although Section 5.5.3 of the AASHTO LRFD BDS specifically excludes deck slabs
from fatigue criterion requirements, this exception presumably applies only to the
190
LRFD Bridge Design
design of the transverse deck reinforcement, given the Commentary note that the
exclusion of decks from fatigue requirements is based on the arching action of the
deck. The behavior of a composite beam is very different from that of a bridge deck,
and such arching is not typically present in bridge girders.
For sections at which the unfactored dead load stress sum is less than that resulting from the Fatigue I (infinite life) limit state, fatigue requirements are satisfied by
limiting the stress range in the deck bars to that given by Equation 7.36, as long as
no welds or splices are present at the section under consideration. The stress, f min,
is the minimum stress resulting from (a) the Fatigue I limit state combined with (b)
the unfactored dead load effects (both DC and DW). Only those DC and DW effects
which act on the composite section, after continuity has been established, should be
included. The stress, f min, is positive if tension, negative if compression.
g ( Df ) = 1.75 ( Df ) £ ( DF )TH = 26 - 22
fmin
fy
(7.36)
If negative moment reinforcement is spliced, then the limiting stress range (ΔF)TH is
reduced significantly to 4.0 ksi.
7.7 TRANSFER AND DEVELOPMENT LENGTH
Transfer length is the distance from the stress-free unbonded end of a strand to the
point at which the strand stress reaches a value equal to the effective prestress, f pe,
after losses.
The calculation of losses in pretensioned members is not covered in this text.
Modern software typically contains multiple options for the calculation of prestress
losses. Refer to Section 5.9.3 of the AASHTO LRFD BDS.
Development length is defined as the distance from the stress-free, unbonded end
of a strand to the point at which strand stress reaches a value equal to the stress at
nominal resistance, f ps.
Transfer length is equal to 60 strand diameters, and development length is equal
to the value given by Equation 7.37. The nominal strand diameter is db. The coefficient, κ, is equal to 1.6 for pretensioned beams with a depth greater than 24 inches,
or 1.0 for other pretensioned members.
2
æ
ö
ld = k ç f ps - f pe ÷ db
3
è
ø
(7.37)
7.8 STRESS CONTROL MEASURES
Controlling stresses in pretensioned bridge girders, both at release and for in-service
conditions, may require one or more of the following:
• draping of strands
• debonding of strands
• increase in the compressive strengths specified for the beam
Precast Prestressed Concrete Girders
191
Draping, also referred to as harping, involves raising strands at the end of the beam
relative to the location at the midspan of the beam in an effort to mitigate overstresses. Draping effectively raises the centroid of the prestressing strands near the
end of the beam. The onset of the drape is typically about 40 percent of the beam
length from each end, and requires special hold-down devices to resist the upward
component of strand force. Figure 7.2 shows the elevation of a prestressed girder
incorporating draped strands to control stresses near the end of the beam. Figure 7.3
shows the mid-span and girder end cross sections for the same girder.
De-bonding involves wrapping strands, typically near the end of the beam, to
prevent bonding to the concrete until such point along the beam as such bonding
does not overstress the beam.
The AASHTO LRFD BDS is valid for concrete strengths up to 15 ksi. Regional
practices may make it difficult to obtain concrete strengths near 15 ksi, and values closer to 10 ksi are often targeted by engineers. Recall that specified strengths
include that at release, f’ci, and that for in-service conditions, f’c. In-service required
strength is typically of the order of 500 to 1,000 psi greater than the required strength
specified at release, though no such range is a requirement.
7.9 SOLVED PROBLEMS
Problem 7.1
For the concrete girder option of the Project Bridge, the 89-foot 3-inch-long
BT-54 girder shown in Figure P7.1 below has been proposed. The section
at midspan is shown in the figure. Prestressing steel includes thirty ½-inchdiameter, 270K low-relaxation strands, arranged as denoted by the solid
circles in the cross-section figure. Deck thickness is 8.25 inches and the
girder spacing is 9 ft 3 inches. Deck concrete f’c is 4.0 ksi. Girder release
and final strengths are f’ci = 7.0 ksi and f’c = 9.2 ksi. The bottom row of
FIGURE 7.2 Partial Girder Elevation Showing Draped Strands.
192
FIGURE 7.3 Cross Sections Showing Draped Strands.
FIGURE P7.1 Problem P7.1.
LRFD Bridge Design
193
Precast Prestressed Concrete Girders
strands is centered 2.5 inches from the bottom of the girder. Subsequent
rows are 2 inches above the previous row. The haunch distance, from the
top of girder to the bottom of deck, is 2 inches. The deck reinforcement over
the intermediate pier is 22 #7 bars (f y = 60 ksi). Assume the centroid of the
22 bars is at mid-thickness of the deck.
Determine the initial pull and the stresses in the girder at release. Compare
the computed stresses to those permitted at release. Use transformed properties and ignore losses due to elastic shortening for stress calculations.
Problem 7.2
Maximum midspan positive (tension in the bottom of the girder) girder
shears and moments for the composite BT-54 in Problem 7.1 are as follows:
case 1 (maximum positive moment)
M DC = 1, 985 ft-kips M DW = 143 ft-kips
VDC = 3 kips
VDW = 5 kips
M LL + IM = 1, 421 ft-kkips
VLL + IM = 33 kips
case 2 (maximum shear)
M DC = 1, 985 ft-kips M DW = 143 ft-kips
M LL + IM = 1,103 ft-kiips
VDC = 3 kips VDW = 5 kips VLL + IM = 44 kips
Determine the required flexural resistance at the Strength limit state, Mu,
and the flexural resistance, ϕMn, in the design provided. The flexural resistance is from the thirty ½-inch-diameter strands in tension and the deck in
compression. Use hand calculations supplemented with a Response 2000
model. Total losses are estimated to be 19.3 ksi. The deck reinforcement is
to be ignored. Shear stirrups are single #6 at 18 inches.
Problem 7.3
Maximum negative (compression in the bottom of the girder) girder
moments and corresponding shears for the composite BT-54 in Problem 7.1
are as follows:
case 1 (maximum negative moment)
M DC = -192 ft-kips M DW = -269 ft-kips M LL + IM = -1, 519 ft-kips
VDC = 94 kips
VDW = 16 kips
VLL + IM = 84 kips
case 2 (maximum shear)
M DC = -192 ft-kips M DW = -269 ft-kips
VDC = 94 kips
VDW = 16 kips
M LL + IM = -612 ft-kipps
VLL + IM = 112 kips
Determine the required flexural resistance at the Strength limit state, Mu,
and the flexural resistance, ϕMn, of the provided design. The flexural resistance provided is carried by the bars in the deck in tension. Stirrups are
double #6 at 6 inches on center extending into the deck.
194
LRFD Bridge Design
Problem 7.4
Estimate the midspan deflection of the BT-54 girder from Problem 7.1. The
effects to be included are prestress and girder self-weight.
Problem 7.5
Determine the continuity requirements for strand extension into the support
diaphragm at the intermediate pier of the two-span Project Bridge.
PROBLEM 7.1
TEH
1/3
195
Precast Prestressed Concrete Girders
PROBLEM 7.1
TEH
2/3
196
PROBLEM 7.1
LRFD Bridge Design
TEH
3/3
197
Precast Prestressed Concrete Girders
PROBLEM 7.2
TEH
Hand calculation ignoring the M–V interaction:
1/4
198
PROBLEM 7.2
LRFD Bridge Design
TEH
2/4
199
Precast Prestressed Concrete Girders
PROBLEM 7.2
With M/V = 75.1 feet:
TEH
3/4
200
PROBLEM 7.2
With M/V = 52.4 feet:
LRFD Bridge Design
TEH
4/4
201
Precast Prestressed Concrete Girders
PROBLEM 7.3
TEH
1/3
202
PROBLEM 7.3
With M/V = −11.43 feet:
LRFD Bridge Design
TEH
2/3
203
Precast Prestressed Concrete Girders
PROBLEM 7.3
With M/V = −5.07 feet:
TEH
3/3
204
PROBLEM 7.4
LRFD Bridge Design
TEH
1/1
205
Precast Prestressed Concrete Girders
PROBLEM 7.5
TEH
1/5
206
PROBLEM 7.5
LRFD Bridge Design
TEH
2/5
207
Precast Prestressed Concrete Girders
PROBLEM 7.5
TEH
3/5
208
PROBLEM 7.5
LRFD Bridge Design
TEH
4/5
209
Precast Prestressed Concrete Girders
PROBLEM 7.5
TEH
5/5
210
LRFD Bridge Design
7.10 EXERCISES
E7.1.
An interior BT-72 girder has forty ½-inch-diameter, 270K low-lax strands
arranged as shown in Figure E7.1. Girder concrete strength f’c = 7,000 psi.
Composite deck thickness is 8.25 inches with a girder spacing S of 11 ft 6
inches. Deck concrete f’c = 4,000 psi. The initial pull on the strands is 75%
of fpu and the estimated total losses are 23.55 ksi. Young’s modulus for the
strands is E = 28,500 ksi. Determine the flexural resistance, ϕMn, in positive bending of the composite girder. Haunch depth is 2 inches. All template
rows shown are 2 inches above the previous row. The bottom row is 2.5
inches from the bottom of the girder.
FIGURE E7.1
Exercise E7.1.
Precast Prestressed Concrete Girders
211
Note: Some engineers do not permit the placement of two strands per
row in a web only 6 inches wide. Such practice, while resulting in a relatively congested geometry, has been used successfully. Consult local fabricators and owners.
E7.2.
The BT-63 girder shown in Figure E7.2 is made from concrete with f’ci =
8 ksi and f’c = 9.5 ksi. Strands are 0.60 inch diameter, 270K low-lax, straight
strands with no de-bonding. Initial pull is 75% of f pu. The girder is made
composite with the 4 ksi concrete deck, which is 8.25 inches thick. Haunch
thickness is 2 inches. The bottom row of strands is 2.5 inches from the
bottom of the girder. Subsequent rows are 2 inches above the previous row.
Initial losses due to elastic shortening are 12.7 ksi. Total final losses are
estimated to be 22.3 ksi.
Strength limit state moment is -4,463 ft-kips with a simultaneous shear
equal to 197 kips. Stirrups (not shown, for clarity) are single #6 spaced at 6
inches on center extending into the deck. The top mat of 16 #7 bars is located
3.7 inches below the top of the deck. The bottom mat of 12 #7 bars is located
2.1 inches from the bottom of the deck. Grade 60 reinforcing is used.
Assess the adequacy of the stirrups and deck longitudinal reinforcement
to carry the required Strength limit state loads. Strands are ineffective at the
girder end and are to be ignored.
FIGURE E7.2 Exercise E7.2.
212
LRFD Bridge Design
Note: Some engineers do not permit the placement of two strands per
row in a web only 6 inches wide. Such practice, while resulting in a relatively congested geometry, has been used successfully. Consult local fabricators and owners.
E7.3.
The BT-63 girder shown in Figure E7.3 is made from concrete with f’ci =
8 ksi and f’c = 9.5 ksi. Strands are 0.60 inch diameter, 270K low-lax, straight
strands with no de-bonding. Initial pull is 75% of f pu. The girder is made
composite with the 4 ksi concrete deck, which is 8.25 inches thick. Haunch
thickness is 2 inches. The bottom row of strands is 2.5 inches from the
bottom of the girder. Subsequent rows are 2 inches above the previous row.
Initial losses due to elastic shortening are 12.7 ksi. Total final losses are
estimated to be 22.3 ksi.
Service I limit state moment is −2,720 ft-kips with simultaneous shear of
175 kips. Stirrups (not shown, for clarity) are single #6 spaced at 6 inches
on center extending into the deck. The top mat of 16 #7 bars is located 3.7
inches below the top of the deck. The bottom mat of 12 #7 bars is located
2.1 inches from the bottom of the deck. Grade 60 reinforcing is used. For
the deck, f’c = 4,000 psi.
FIGURE E7.3
Exercise E7.3.
Precast Prestressed Concrete Girders
213
Assess the adequacy of the deck reinforcing to meet crack control
requirements at the Service limit state. Strands are ineffective at the girder
end and are to be ignored.
Note: Some engineers do not permit the placement of two strands per
row in a web only 6 inches wide. Such practice, while resulting in a relatively congested geometry, has been used successfully. Consult local fabricators and owners.
E7.4.
The BT-63 girder shown in Figure E7.4 is made from concrete with f’ci =
8 ksi and f’c = 9.5 ksi. Strands are 0.60 inch diameter, 270K low-lax, straight
strands with no de-bonding. Initial pull is 75% of f pu. The girder is made
composite with the 4 ksi concrete deck, which is 8.25 inches thick. Haunch
thickness is 2 inches. The bottom row of strands is 2.5 inches from the
bottom of the girder. Subsequent rows are 2 inches above the previous row.
Initial losses due to elastic shortening are 12.7 ksi. Total final losses are
estimated to be 22.3 ksi.
Strength limit state moment, Mu, is +7,208 ft-kips at a distance 40 percent
of the span length from the abutment (the point of maximum positive moment).
The shear, Vu, acting simultaneously with the moment, is 46 kips. Grade 60
reinforcing is used. Stirrups (not shown, for clarity) are single #6 at 18 inches.
FIGURE E7.4 Exercise E7.4.
214
LRFD Bridge Design
Determine the design flexural and shear resistances, ϕMn and ϕVn, and
compare with the required values, Mu and Vu.
Note: Some engineers do not permit the placement of twi strands per row
in a web only 6 inches wide. Such practice, while resulting in a relatively
congested geometry, has been used successfully. Consult local fabricators
and owners.
E7.5.
For the girder of exercise E7.4, estimate midspan stresses and deflections
at release. Compare the stresses to permissible values. The girder length is
113 ft.
E7.6.
A 125-ft-long BT-72 bridge girder has forty 0.60-inch-diameter, 270K lowlax strands, 14 of which are draped at 0.4L. The strand group is raised a
total of 4 ft 4 inches from the drape point to the girder end. Initial pull is 75
percent of fpu. Determine (a) the required hold-down force at the drape point
for fabrication of the girder and (b) the stress in the straight portion of the
draped strands under initial pull.
E7.7.
For the girder of exercise E7.4, determine the required non-prestressed mild
reinforcement in the top of the girder to resist tension prior to losses.
E7.8.
For the girder in exercise E7.3, determine the required number and length
of bent strands at the girder ends to provide the required positive moment
connection. Assume that the girder age at continuity is no less than 90 days
and thus, restraint moments may be ignored in the design.
8
Bridge Girder Bearings
Two popular bearing configurations are (a) elastomeric bearing pads, plain or reinforced with steel layers, and (b) steel assemblies, as either fixed or expansion bearings. Elastomeric pads may also be constructed to behave as either fixed or expansion
bearings.
Each of these bearing types, with modification, may be used as seismic isolation
bearings as well.
Other bearing types include metal roller systems, PTFE sliding surface systems,
pot bearings, and disc bearings. These are not covered here, and Chapter 14 of the
AASHTO LRFD BDS contains detailed requirements for such.
8.1 ELASTOMERIC BEARINGS
This section summarizes design criteria for elastomeric bearings in the AASHTO
LRFD BDS. The reader is referred to Section 14.7.5 of the AASHTO LRFD BDS for
the basis of the material presented here, and for further explanations and discussion
on the design of elastomeric bearings for bridge girders.
Figure 8.1 depicts details for an expansion elastomeric bearing used in a bridge in
Clay County, Tennessee on State Route 52.
Prior to discussion of design criteria, a definition of terms will be helpful.
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
hrt = total elastomer thickness, exclusive of steel layers
hri = thickness of ith elastomer layer
G = specified shear modulus of elastomer, ksi
ΔS = service shear deformation
Si = shape factor
L = plan dimension of a rectangular bearing perpendicular to the axis of
rotation
W = plan dimension of a rectangular bearing parallel to the axis of rotation
D = diameter of a round bearing
d = diameter of a hole through an elastomeric bearing
Dr = dimensionless rotation coefficient (0.500 for rectangular bearings,
0.375 for circular bearings)
Da = dimensionless axial coefficient (1.4 for rectangular bearings, 1.0 for
circular bearings)
γa = shear strain caused by axial load
γr = shear strain caused by rotation
γs = shear strain caused by shear displacement
γa,st = shear strain due to non-traffic axial load
DOI: 10.1201/9781003265467-8
215
216
LRFD Bridge Design
FIGURE 8.1 State Route 52 over branch – elastomeric bearing details.
•
•
•
•
•
•
•
•
•
•
•
•
•
γa,cy = shear strain due to traffic-induced axial load
γr,st = shear strain due to non-traffic rotation
γr,cy = shear strain due to traffic-induced rotation
γs,st = shear strain due to non-traffic shear
γs,cy = shear strain due to traffic-induced shear
σs = Service limit state total compressive stress on the bearing (all load
factors = 1.00)
σL = Service limit state live load compressive stress on the bearing (load
factor = 1.00)
σhyd = peak hydrostatic stress
θs = service limit state rotation; to include an additional 0.005 radians for
miscellaneous uncertainties
hs = thickness of steel reinforcement layers
n = number of interior layers of elastomer; when the thickness of the exterior
layer of elastomer is equal to or greater than one-half the thickness of an
interior layer, the parameter, n, may be increased by one-half for each such
exterior layer.
Fy = yield stress of steel layers
ΔFTH = Category ‘A’ fatigue threshold (24 ksi)
The cover layer thickness of a reinforced elastomeric bearing is to be no more than
70% of the internal layer thickness. All internal elastomer layers shall be the same
thickness.
The shape factor, Si, of an elastomer layer is defined as the plan area of the
layer divided by the area of perimeter free to bulge. The shape factor is defined
in Equation 8.1 for rectangular bearings, or in Equation 8.2 for round bearings.
Although the equations do allow for the presence of holes in the elastomeric bearing
217
Bridge Girder Bearings
to accommodate anchor rods, such practice is generally to be discouraged. It is usually possible to place the anchor rods outside the limits of the elastomeric bearing.
p d2
4
, rectangular
Si =
hri éë2 L + 2W + å p d ùû
LW - å
Si =
D2 - å d 2
, round
4hri ( D + å d )
(8.1)
(8.2)
In addition to plan dimensions and elastomer and steel layer details, elastomeric
bearing definition and design must include a specified shear modulus, G. AASHTO
requires that (a) the shear modulus specified on the plans must be in the range of
0.080 to 0.175 ksi, (b) a tolerance of ±15% on the specified shear modulus is to be
incorporated into the design, and (c) the least favorable value is to be assigned to the
shear modulus used in calculations. Generally speaking, the minimal G (0.85 times
the specified value) within the range will be the least favorable for stress checks and
the maximal G (1.15 times the specified value) within the range will be the least
favorable in rotation capacity calculations or in determining the loads delivered to
the substructures by the bearings.
Not only a shear modulus, G, but also a minimum low-temperature grade, should
be specified for elastomeric material used in reinforced elastomeric bearings.
Elastomeric materials stiffen as temperature decreases. In Section 14.7.5.2 of the
AASHTO LRFD BDS, five temperature zones are mapped, with Zone A being the
least severe and Zone E the most severe. The minimum low-temperature grade of
elastomer suitable for each zone, with forces delivered to substructures as presented
later, is summarized below. It is permissible to use a lower-than-recommended grade,
but with increased forces delivered to the substructures for design. Refer to Section
14.7.5.2 of the AASHTO LRFD BDS for such provisions.
•
•
•
•
•
Low-temperature Zone A, minimum low-temperature grade = 0
Low-temperature Zone B, minimum low-temperature grade = 2
Low-temperature Zone C, minimum low-temperature grade = 3
Low-temperature Zone D, minimum low-temperature grade = 4
Low-temperature Zone E, minimum low-temperature grade = 5
For thermal expansion and contraction, the design movement is taken to be equal
to 65% of the total thermal movement range to account for variation in installation
temperatures.
Expansion bearings are subject to shear displacement, whereas fixed bearings are
not. For expansion bearings, the Service limit state shear deformation is limited to
that given by Equation 8.3.
Ds £
hrt
2
(8.3)
218
LRFD Bridge Design
The minimum thickness for steel layers is 0.0625 inches. Steel layers must satisfy
requirements at both the Service and Fatigue limit states, as defined by Equations
8.4 and 8.5, respectively.
hs ³
3hris s
Fy
(8.4)
hs ³
2hris L
DFTH
(8.5)
Stability checks for elastomeric bearings are given in Equation 8.6, for expansion
bearings, and Equation 8.7 for fixed bearings. Stability checks for circular bearings
are to be made using L = W = 0.8D. If 2A ≤ B, then the bearing is stable and the stress
checks given by the Equations need not be made. Otherwise, the stress check must
be satisfied to ensure stability of the bearing.
ss £
GSi
2A - B
(8.6)
ss £
GSi
A-B
(8.7)
hrt
L
A=
L
1+ 2
W
(8.8)
1.92
B=
2.67
L
( Si + 2.0 ) æç 1 + 4W ö÷
è
ø
(8.9)
Shear strains for bearing design are determined using Equations 8.10 through 8.13,
with the limiting values provided in Equations 8.14 and 8.15.
g a = Da
ss
GSi
(8.10)
2
æ L ö q
g r = Dr ç ÷ s , rectangular
è hri ø n
(8.11)
2
æDö q
g r = Dr ç ÷ s , circular
è hri ø n
(8.12)
219
Bridge Girder Bearings
gs =
Ds
hrt
(8.13)
g a,st £ 3.0
(8.14)
(g a,st + g r ,st + g s,st ) + 1.75 (g a,cy + g r ,cy + g s,cy ) £ 5.0
(8.15)
For bearings with bonded plates on both top and bottom, the additional requirements
given in Equations 8.16 through 8.22 must be satisfied. The coefficient, Ba, is taken to
be equal to 1.6 unless a more refined solution can be justified. For bearings without
externally bonded plates, a restraint system to prevent lateral movement is required
whenever Equation 8.23 is found to be valid.
s hyd = 3GSi3
qs2
Ca
n
(8.16)
1.5
ù
4 éæ
1ö
Ca = êç a 2 + ÷ - a 1 - a 2 ú
3 êëè
3ø
úû
(
)
(8.17)
a=
ea n
×
Si q s 2
(8.18)
ea =
s s2
3BaGSi2
(8.19)
s hyd £ 2.25G
(8.20)
PD + 1.75PL
A
(8.21)
q s 2 = q D + 1.75q L
(8.22)
q s 3e a
³
n
Si
(8.23)
s s2 =
The elastomeric bearing, once final design dimensions and properties have been
established to satisfy rotational, strain, stress, and stability checks, must be analyzed
to determine the magnitude of forces delivered to the substructure by the bearing.
Equations 8.24 and 8.25 provide a means of determining both the shear force and the
moment delivered to the supporting member by the bearing.
220
LRFD Bridge Design
H bu = GA
Du
hrt
Mu =1.60 ( 0.5Ec I )
I=
(8.24)
qs
hrt
(8.25)
Ec = 4.8GSi2
(8.26)
WL3
, rectangular
12
(8.27)
p D4
, circular
64
(8.28)
I=
8.2 STEEL ASSEMBLY BEARINGS
For large vertical reactions and large thermal movement requirements, it may be
necessary to use steel assembly bearings. Figure 8.2 shows an elevation at the girder
end for the expansion bearing used for a bridge over Center Hill Lake in Smithville
Tennessee on State Route 26. Figure 8.3 shows the fixed counterpart for the Center
Hill Lake Bridge. Fixed bearings were used at the west abutment and at the four
piers for the 1,545-foot-long steel girder bridge. Expansion bearings were used for
the girders at the east abutment, where the entire thermal expansion and contraction
requirements were accommodated.
FIGURE 8.2 Expansion bearing for State Route 26 over Center Hill Lake, Smithville,
Tennessee.
221
Bridge Girder Bearings
FIGURE 8.3
Fixed bearing for State Route 26 over Center Hill Lake, Smithville, Tennessee.
Observe, from Figure 8.2, that the total movement capacity of the bearing is equal
to SW−BW, before the restrainers engage.
Either bronze or copper slider plates, at the contractor’s option, were permitted in
the contract documents for the bearings shown in Figure 8.2. The AASHTO LRFD
BDS, in Section 14.7.7.2, specifies a coefficient of friction equal to 0.10 for selflubricating bronze and 0.40 for other slider plate materials. Frictional forces (FR)
acting laterally for either expansion or contraction must be accommodated in the
substructure design. Such forces would need to be based on the frictional forces
developed with the higher coefficient.
Bearing stress at the Service limit state is limited to 2.0 ksi for Type 1 and 2
bronze plates in the AASHTO LRFD BDS, Section 14.7.7.3. Bearing stress on concrete at the Strength limit state is limited to that given by Equation 8.29, where A1
is the plan area of the bearing contact surface. See Section 5.6.5 of the AASHTO
LRFD BDS. Bearing pressure checks on both the bronze/copper slider plate and on
the concrete support should be made under the fully extended condition for expansion bearings. The coefficient, m, may conservatively be taken to be equal to 1.0.
When the supporting concrete is wider than the bearing on all sides, m may reach
values as high as 2.0. Refer to Section 5.6.5 of the AASHTO LRFD BDS for methods which may be used to justify m-values larger than 1.0.
f Pn = 0.70 éë0.85 fc¢A1m ùû
(8.29)
222
LRFD Bridge Design
8.3 ISOLATION BEARINGS
Seismic isolation bearings include lead-rubber bearings (LRB) and friction pendulum systems (FPS), among others.
Figure 8.4 shows an elevation of LRB bearings used at the piers for Interstate 40
over State Route 5 in Madison County, Tennessee. The bearings are 22.5 inches in
diameter with 3.75-inch-diameter lead plug for energy dissipation. Non-seismic displacement for the bearings is 0.52 inches and the total designed displacement during
a seismic event is 8.30 inches.
Figure 8.5 is a schematic diagram of the FPS isolator bearings used in the retrofit
of the main piers for Interstate 40 over the Mississippi River in Memphis, Tennessee
(Figure 8.6). The bearings are 8 ft 10inches in diameter with a concave surface
radius of 244 inches. The FPS system used for this bridge is capable of accommodating up to 27.25 inches of seismic displacement with a maximum vertical load
capacity of 12,611 kips.
For LRB isolators, the parameters defining behavior are summarized as follows:
• f
• G
• Gp
factor to account for post-yield stiffness of the lead core (1.1, typical values)
elastomer shear modulus (50–300 psi, typical values)
shear modulus of the lead plug (21.75 ksi, typical values)
FIGURE 8.4 Interstate 40 over State Route 5 isolation bearings.
Bridge Girder Bearings
223
FIGURE 8.5 Friction pendulum system (FPS) bearings for the Hernando de Soto Bridge in
Memphis, Tennessee.
•
•
•
•
•
•
•
•
•
•
•
•
Tr total rubber (elastomer) thickness
Ab bonded area of rubber
db diameter of circular bearing
dL diameter of lead plug (db/6 to db/3, typical values)
f yL yield stress in shear for lead (1.3–1.5 ksi, typical values)
ψ
stress modifier for lead plug:
• 1 for EQ-load
• 2 for wind/braking
• 3 for thermal expansion
α
post-yield stiffness ratio, typical values:
• 0.10 for EQ
• 0.125 for wind/braking
• 0.20 for thermal expansion
γc shear strain due to compression
γr shear strain due to rotation
γs,s shear strain due to non-seismic displacement
γs,eq shear strain due to seismic displacement
ti
thickness of an individual elastomer layer
FIGURE 8.6
Hernando de Soto Bridge in Memphis, Tennessee.
224
LRFD Bridge Design
225
Bridge Girder Bearings
θ
bearing rotation (include a 0.005 radian contingency)
B bonded plan dimension in the direction of loading (db for circular)
ξEFF effective damping imparted into a system through inelastic behavior
BL response modification divisor to account for added damping
Qd characteristic strength
kd post-yield stiffness
Δs lateral non-seismic displacement demand on an isolation bearing
DISO lateral seismic displacement demand on an isolation bearing
S
shape factor; bonded plan area divided by the side area free to bulge
Dr shear strain factor
• 0.375, circular bearing
• 0.500, rectangular bearing
• DC shape coefficient
• 1.000, circular bearing
• 1.000, rectangular bearing
• σs compressive stress = P/Ab
•
•
•
•
•
•
•
•
•
•
Design criteria for LRB isolators used in bridge structures may be found in AASHTO
GS ISO design specifications (AASHTO, 2014), supplemented with literature on the
subject (Stafford et al., 2008; Warn, 2002). A subset of applicable criteria is summarized in Equations 8.30 through 8.46. Loads that contribute to the static components
of deformation include wind, dead load, and thermal effects. Loads that are assumed
to contribute to the cyclic components of deformation include live load, braking
forces, and seismic effects.
Equations for these systems are typically approximate and final properties are
generally established through rigorous testing of the bearings.
Ab =
(
p db2 - dL2
)
4
kd = f
GAb
Tr
(8.30)
(8.31)
ki =
G p Ap + GAb
Tr
(8.32)
Fy =
p d2
1
× f yL × L
y
4
(8.33)
db2 - dL2
4 db t i
(8.34)
Dcs s
GS
(8.35)
S=
gc =
226
LRFD Bridge Design
gr =
Dr B2q
tiTr
g s,s = D s
Tr
g s,eq = DISO
Tr
(8.36)
(8.37)
(8.38)
(g c + g r + g s,s )static + 1.75 (g c + g r + g s,s )cyclic £ 5.0
(8.39)
(g c )static £ 3.0
(8.40)
(g c + 0.50g r + g s,s + g s,eq )total £ 5.5
(8.41)
K EFF = kd +
x EFF =
Qd
DISO
2Qd ( DISO - Dy )
p ( DISO ) K EFF
2
Dy =
(8.42)
(8.43)
Qd a
×
kd 1 - a
(8.44)
Qd
1-a
(8.45)
Fy =
æx
ö
BL = ç EFF ÷
.
0
05
è
ø
0.30
(8.46)
Design parameters for friction pendulum systems (FPS) include:
•
•
•
•
μ
R
W
Dvert
dynamic friction coefficient (0.03–0.12, typical values)
radius of concave surface
vertical load
vertical displacement due to concave sliding surface
A summary of relationships for FPS isolation systems is provided in Equations 8.47
through 8.53.
Qd = mW
(8.47)
227
Bridge Girder Bearings
W
R
(8.48)
W
W
+m×
R
DISO
(8.49)
m
2
×
p m + DISO / R
(8.50)
kd =
keff =
x eff =
j = sin -1
DISO
R
Dvert = R (1 - cos j ) @
m£
(8.51)
2
DISO
2R
DISO
£ 0.15
R
(8.52)
(8.53)
Although not a specification requirement, limits on DISO/R given in Equation 8.53
are generally considered to be good practice. The lower limit establishes improved
re-centering capability. The upper limit assures that the small rotation angle assumption about the center of curvature of the concave surface is valid. As with LRB
devices, the equations for FPS systems are approximate and final properties are
established through testing.
8.4 ANCHOR RODS
The design shear resistance, ϕRn, for ASTM F1554 anchor rods in shear is given by
Equation 8.54, where Fub is the specified minimum tensile strength of the anchor
rod, Ab is the nominal area of the anchor rod, and Ns is the number of shear planes,
typically equal to 1 for anchored bearings. Refer to Section 6.13.2.12 of the LRFD
BDS for additional information. Chapter 1 of this text includes a summary of the
mechanical properties and available diameters for ASTM F 1554 anchor rods.
f Pn = 0.75 éë0.50 Fub Ab N s ùû
(8.54)
Anchor rods are commonly used to carry lateral forces from the bearings to the pier
cap in both steel girder and concrete girder bridges. These anchors may be located
at the bearings, embedded in pier diaphragms, or both. The number of shear planes,
Ns, for such anchors is one.
8.5 SOLVED PROBLEMS
Problem 8.1
A prestressed concrete beam bridge incorporates the bearing design shown
in the end elevation (Figure P8.1). Dead load rotation may be taken as zero.
228
LRFD Bridge Design
FIGURE P8.1 Problem P8.1.
Live load rotation is 0.009 radians. Miscellaneous static rotation is 0.005
radians. Axial loads are 425 kips (DL) and 275 kips (LL). The anticipated
thermal movement is 2.00 inches each way. The specified shear modulus is
175 psi. AASHTO requires that a ± 15% deviation in specified shear modulus be incorporated into all design checks, using the least favorable value in
the range.
Check the bearing for the following AASHTO requirements:
a) shear deformation limits
b) strain limits
c) stability requirements
d) steel reinforcement requirements
e) dimensional requirements
f) determination of the shear and moment the bearing delivers to the
substructure
Problem 8.2
For the Sligo bridge expansion bearings, determine the required dimensions SW and BW. Establish the maximum design resistance for the bearing
reaction based (a) on the slider plate and (b) on the concrete cap. The bridge
is a steel girder bridge in a moderate climate. The entire expansion requirement for the 1,545-ft-long bridge is to be taken by expansion bearings at
Abutment 2. Refer to Figures 8.2 and 8.3.
The girder reactions are:
RDC = 236 kips RDW = 48 kips RLL + IM = 167 kips
Bridge Girder Bearings
229
The baseplate area is BL ×BW. The slider plate area is CW ×L. Take:
• CW = BW–2
• BL = 36 inches
• CL = 28 inches
Problem 8.3
An anchor rod made from ASTM F 1554 material is to be used to carry
seismic shear forces from the superstructure to the intermediate pier for a
two-span bridge. Span lengths are 140 ft each and the bridge is 45 ft wide.
Five girders are spaced 9 ft 6 inches apart on center. The total superstructure weight is 13.1 kips per foot, including two lanes of HL-93 uniform
lane loading. The anchors are embedded in a 1-ft 6-inch-wide diaphragm
at the pier. At the project site, SDS = 0.854 g and SD1 = 0.733 g. The fundamental period of the structure is 0.95 seconds. For preliminary design, a
simplified analysis is to be used for sizing the anchors. The elastic seismic
force is to be reduced by a factor of 1.5 for the simplified analysis. Final
design will ensure that the anchors can resist the overstrength plastic shear
of the pier. Design the preliminary anchor configuration (by determining
the anchor grade, the anchor diameter, and the number of required anchors
at the pier).
Problem 8.4
A 31.5-inch-diameter isolation bearing (LRB) consists of 15 internal elastomer layers, each 0.75 inches thick. External layers are each 0.375 inches
thick and the internal steel layers (16) are each 0.125 inches thick. The
elastomer shear modulus is G = 55 psi. Shear yield stress for the 8-inchdiameter lead plug is 1.3 ksi. The shear modulus for the lead plug is
G = 21,750 psi. The maximum seismic displacement is estimated to be 20
inches. Service level rotation is 0.010 radians (100% cyclic) and deflections due to non-seismic loads are 3.8 inches (TU) and 0.7 inches (BR).
Determine:
• the initial stiffness and yielded stiffness for the bearing
• the yield force and yield displacement for the bearing
• the effective damping at maximum seismic displacement
• the force delivered to the substructure by the bearing at maximum seismic displacement.
• The maximum compressive force to be permitted on the bearing if the
ratio of total dead load to total live load (D/L) is 1.68).
230
PROBLEM 8.1
LRFD Bridge Design
TEH
1/3
231
Bridge Girder Bearings
PROBLEM 8.1
TEH
2/3
232
PROBLEM 8.1
LRFD Bridge Design
TEH
3/3
233
Bridge Girder Bearings
PROBLEM 8.2
TEH
1/2
234
PROBLEM 8.2
LRFD Bridge Design
TEH
2/2
235
Bridge Girder Bearings
PROBLEM 8.3
TEH
1/1
236
PROBLEM 8.4
LRFD Bridge Design
TEH
1/3
237
Bridge Girder Bearings
PROBLEM 8.4
TEH
2/3
238
PROBLEM 8.4
LRFD Bridge Design
TEH
3/3
Bridge Girder Bearings
239
8.6 EXERCISES
E8.1.
For the Project Bridge, estimated reactions for an interior girder are as
follows:
PDC1 = 142 kips per girder
PDC2 = 23 kips per girder
PDW = 32 kips per girder
PLL+IM = 180 kips per girder
A fixed bearing (no expansion/contraction movement) is proposed at the
pier. A reinforced elastomeric bearing with specified shear modulus G = 100
psi is proposed. Bearing dimensions are L = 12 inches and W = 16 inches.
The proposed bearing has three internal elastomer layers, each 0.50 inches
thick, and two external elastomer layers, each 0.25 inches thick. Internal
reinforcement layers are 0.125-inch-thick A36 steel. The bearing has no
holes and bonded steel plate on top only. Live load rotation is 0.002 radians.
Dead load rotation is taken to be zero.
Check the strain and stability requirements, check the steel layer thickness, and determine the forces delivered to the pier.
E8.2.
A 30-inch (W) × 18‑inch (L) reinforced elastomeric expansion bearing is
proposed for a bridge girder with the following loads:
PDC = 200 kips
PDW = 55 kips
PLL+IM = 160 kips
θLL = 0.0035 radians
Δ = 3.50 inches each way
The bearing is to consist of ½-inch-thick internal elastomer layers with
¼-inch-thick external cover layers and 1/8-inch-thick steel reinforcement
layers. Specified elastomer shear modulus is 100 psi. Dead load rotation is
accommodated by using beveled sole plates and may be taken to be equal
to zero for the design. Determine the minimum number of internal layers
based on expansion requirements. Check the bearing for all other criteria.
9
Reinforced Concrete
Substructures
Many bridge substructures today are constructed using reinforced concrete. These
include bent caps and columns, spread footings, pile caps, drilled shafts, and micropiles. Whereas accelerated bridge construction measures have incorporated precast
components, the material presented here for conventionally constructed substructures focuses on cast-in-place components, with the exception of driven concrete
piles.
Load and resistance factors for buildings and other structures are not necessarily
appropriate for bridge substructures. Load factors for bridges have been discussed in
early chapters of this book. A sampling of AASHTO LRFD BDS resistance factors
for reinforced concrete elements is summarized in Table 9.1.
Tension-controlled reinforced concrete sections are defined as sections with a net
tensile strain in the extreme layer of tensile reinforcement, εt, greater than or equal
to the tension-controlled strain limit, εtl, when the concrete strain reaches a value of
0.003.
Compression-controlled reinforced concrete sections are defined as sections with
a net tensile strain in the extreme layer of tensile reinforcement, εt, less than or equal
to the compression-controlled strain limit, εcl, when the concrete strain reaches a
value of 0.003.
The tension-controlled strain limit, εtl, is determined as follows:
• εtl = 0.005 for reinforcement with f y ≤ 75 ksi
• εtl = 0.008 for reinforcement with f y = 100 ksi
• εtl is determined by linear interpolation for reinforcement with 75 < f y < 100 ksi
The compression-controlled strain limit, εcl, is determined as follows:
• εcl = f y/Es, but not > 0.002, for reinforcement with f y ≤ 60 ksi
• εcl = 0.004 for reinforcement with f y = 100 ksi
• εcl is determined by linear interpolation for reinforcement with 60 < fy < 100 ksi
For sections with net tensile strain in the extreme layer of tension reinforcement
between εcl and εtl, linear interpolation, as given by Equation 9.1, is used to determine the appropriate resistance factor.
0.75 £ f = 0.75 +
DOI: 10.1201/9781003265467-9
0.15 ( e t - e cl )
£ 0.90
e tl - e cl
(9.1)
241
242
LRFD Bridge Design
TABLE 9.1
AASHTO Resistance Factors for Concrete Elements
Resistance factor, ϕ
Condition
Tension-controlled reinforced concrete
0.90
Compression-controlled reinforced concrete
Shear and torsion
Bearing on concrete
0.75
0.90
0.70
Resistance during pile driving
1.00
This chapter provides a discussion of design considerations for pier caps, pier columns, spread footings, pile caps, drilled shafts, and pile bents.
9.1 PIER CAP DESIGN
When the depth of a reinforced concrete flexural member, dl, is greater than 3 feet,
side bars are required. Typical pier caps exceed 3 feet in depth, so side steel, in accordance with Equation 9.2, is often required. See the AASHTO LRFD BDS, Section
5.6.7. This reinforcement is effective in controlling crack widths. The depth, dl, is
defined as the distance from the extreme compression fiber to the centroid of the
extreme steel tensile layer. The side bars, having area on each side face equal to Ask
(in2/ft), are to be located in each side face for a distance of no less than dl/2 from the
tension bars, and at a spacing to exceed neither dl/6 nor 12 inches. The total area of
side bars must not exceed 25% of the total tensile reinforcement area.
Ask ³ 0.012 ( dl - 30 )
(9.2)
Also, to control cracks, Section 5.6 of the AASHTO LRFD BDS requires that bar
spacing in the extreme tension layer of reinforcement must not exceed s as given by
Equation 9.3.
s£
700g e
- 2 dc
b s fss
bs = 1 +
dc
0.7 ( h - dc )
(9.3)
(9.4)
βs is the ratio of strain at the extreme tension face to strain in the centroid of the reinforcement layer closest to the tension face. dc is the distance from the extreme tension
fiber to the centroid of the reinforcement closest to the tension face. The overall height
of the member is h. The Service limit state stress in the reinforcement is fss, which is
not to exceed 0.60 times f y. The exposure factor, γe, is 1.00 for Class 1 exposure conditions and 0.75 for Class 2 exposure conditions. Class 1 exposure corresponds to an
243
Reinforced Concrete Substructures
estimated crack width of 0.017 inches. Class 2 exposure corresponds to an estimated
crack width of 0.013 inches, as the exposure factor is directly proportional to the estimated crack width. Decks and substructures exposed to water are examples of cases
where it may be advisable to use Class 2 exposure for crack control.
The summary of shear provisions in reinforced concrete members presented here
is applicable to so-called “B-regions” without significant torsion. These are regions
where the assumption that plane sections remain plane is judged to be valid. For deep
beams and other disturbed regions (“D-regions”) the reader is referred to Section 5.8
of the AASHTO LRFD BDS.
The shear depth, dv, is taken to be equal to the distance between cross-sectional
resultant tensile and compressive forces but must not be less than either 0.72h or
0.90de. de is the distance from the extreme compression fiber to the resultant crosssectional tensile force and h is the total member depth. The shear width, bv, is the
minimum web width.
For circular members, dv may be determined by cross-sectional analysis, or
Equation 9.5 from the LRFD BDS Commentary to Section 5.7.2.8 may be used. The
column diameter, D, and the diameter of the circle passing through the centroid of
the longitudinal reinforcement, Dr, are readily available prior to any design having
been completed.
æD D ö
dv = 0.9de = 0.9 ç + r ÷
(9.5)
è2 p ø
Nominal shear resistance in non-prestressed members, Vn, consists of contributions
from the concrete and from the steel shear reinforcement. Equation 9.6 provides the
nominal resistance. For normal-weight concrete with stirrups inclined 90 degrees
to the longitudinal axis of the beam, Equations 9.7 and 9.8 provide the concrete and
steel contributions, respectively. The units on f’c must be ksi.
Vn = Vc + Vs £ 0.25 fc¢bv dv
(9.6)
Vc = 0.0316 b fc¢bv dv
(9.7)
Vs =
Av f y dv
× cot q
s
(9.8)
The minimum area of shear reinforcement in non-prestressed members, required
whenever Vu > 0.5ϕ(Vc), is given by Equation 9.9 for normal-weight concrete. The
resistance factor for shear is 0.90.
Av ³ 0.0316 fc¢ ×
bv s
fy
(9.9)
Shear reinforcement spacing limits depend on the shear stress on the concrete, vu,
which is calculated using Equation 9.10 for non-prestressed members. The spacing
limit is given in Equation 9.11.
244
LRFD Bridge Design
vu =
Vu
f bv dv
(9.10)
ì 0.4dv £ 12 inches, if vu ³ 0.125 fc¢
smax = í
î0.8dv £ 24 inches, if vu < 0.125 fc¢
(9.11)
Determination of the coefficients β and θ is necessary to accurately assess shear
resistance. These factors depend on the net tensile strain in the centroid of the tension reinforcement, εs. The discussion here is limited to sections with at least the
minimum amount of transverse shear reinforcement. For such cases, the strain and
the coefficients are given by Equations 9.12 through 9.14 for non-prestressed members. The steel area, As, is the area of longitudinal reinforcement on the flexural tension side of the member only. The strain may also be determined by a cross-sectional
analysis. For other cases, the reader is referred to Section 5.7.3.4.2 of the AASHTO
LRFD BDS.
æ Mu
+ 0.5N u + Vu
çç
dv
es = è
Es As
b=
4.8
1 + 750e s
q = 29 + 3500e s
ö
÷÷
ø
(9.12)
(9.13)
(9.14)
For calculated strains, εs, less than zero, εs may be taken to be equal to zero. The upper
limit on εs to be used for calculation of shear resistance coefficients is εs ≤ 0.006.
This effectively places a lower limit on β equal to 0.873 and an upper limit on θ equal
to 50 degrees. Other limitations on the parameters are summarized below.
• Nu, the factored axial force on the cross section, is positive if tension, negative if compression.
• Mu, the factored moment at the section, must not be less than Vu × dv.
• For sections closer than dv to the face of support, εs at a distance dv from
the face of support may be used to determine β and θ, unless a concentrated
load is located within dv from the support, in which case εs should be determined at the face of support.
For members with small Mu/Vu ratios, relative to member depth, it may be advisable to use detailed, complex section analysis, including flexure-shear interaction.
Problem 9.5 demonstrates the significant difference such an analysis may make in
estimating design resistance.
For additional shear-related longitudinal reinforcement requirements, refer to
Section 5.7.3.5 of the AASHTO LRFD BDS.
245
Reinforced Concrete Substructures
9.2 PIER COLUMN DESIGN
Column reinforcement must typically be in the range of 1% to 8% of the gross column area. Equation 9.15 provides the minimum steel area, irrespective of any seismic design considerations, from the LRFD BDS, Section 5.6.4.2.
æ f¢ ö
As ³ 0.135 Ag ç c ÷
è fy ø
(9.15)
However, for Seismic Zone 2, column longitudinal reinforcement must be between
1% and 6% of the gross cross-sectional area. For Seismic Zones 3 and 4, the longitudinal column reinforcement must be between 1% and 4% of the gross concrete area.
For columns not considered braced against sidesway, slenderness effects may be
ignored whenever Klu/r is less than 22. For columns considered braced against sidesway, slenderness may be ignored whenever Klu/r is less than 34-12 (M1/M2). M1 and
M2 are the smaller and larger end moments, respectively, and the ratio of the two is
positive when the column is in single curvature, negative if the column is in double
curvature. Approximate second-order analysis is acceptable for the evaluation of
slenderness effects whenever Klu/r is less than 100. Equation 9.16 is an acceptable
estimate for the flexural rigidity, EI. The ratio of factored permanent moment to
factored total moment, βd, appears in the equation and requires a measure of engineering judgment. Equation 9.17 provides the critical buckling load. Equations 9.18
and 9.19 give the required amplifiers on column moments producing no appreciable
sidesway (δb) and on column moments resulting in appreciable sidesway (δs).
EI =
Ec I g
2.5 (1 + b d )
Pe =
db =
ds =
p 2 EI
( Klu )
2
1
Pu
10.75Pe
1
å Pu
10.75 å Pe
Mu = d b Mub + d s Mus
(9.16)
(9.17)
(9.18)
(9.19)
(9.20)
Factored axial resistance of non-prestressed concrete columns, ϕPn, may be determined from Equation 9.21 for columns with spiral reinforcement, or Equation 9.22
for tied columns. The factor, kc, is equal to 0.85 for concrete strengths up to and
246
LRFD Bridge Design
including 10 ksi, 0.75 for concrete strengths 15 ksi or more, and linear interpolation
is used for intermediate concrete strength values.
f Pn = f ´ 0.85 éë kc fc¢ ( Ag - Ast ) + f y Ast ùû
(9.21)
f Pn = f ´ 0.80 éë kc fc¢ ( Ag - Ast ) + f y Ast ùû
(9.22)
Biaxial flexure due to eccentricity of axial load effects may be determined from single-axis eccentricities using the following formulas. Equation 9.23 is applicable whenever the factored axial load, Pu, is less than 0.10f’cAg. Equation 9.24 applies otherwise.
M
Mux
+ uy £ 1.00
f M nx f M ny
(9.23)
1
1
1
1
=
+
+
f Pnxy f Pnx f Pny f Po
(9.24)
f Po = f éë kc fc¢ ( Ag - Ast ) + f y Ast ùû
(9.25)
For circular columns of diameter D, and with a reinforcing ring of diameter Dr, the
effective shear depth, dv, may be determined from Equation 9.26.
æD D ö
dv = 0.90 ç + r ÷
è2 p ø
(9.26)
Column design generally requires criterion checks at both the minimum and maximum axial load levels, and at multiple Strength limit states.
9.3 SPREAD FOOTING DESIGN
Footing design provisions in the AASHTO LRFD BDS are found in Chapter 5
(Section 5.12.8) and Chapter 10 (Section 10.6).
For square footings, the LRFD BDS in Section 5.12.8 requires footing reinforcement to be uniformly distributed across the footing width in each direction.
For rectangular footings, long bars are to be distributed uniformly across the short
dimension, and short bars are to be distributed along the long dimension as follows.
A central band, equal in width to the short dimension of the footing, is to have a
uniformly distributed area of steel equal to As-BW. The remainder of the required steel
is to be uniformly distributed on each side of the central band. The total steel area
required by analysis for the short bars is denoted As-SD. Equation 9.27 gives the area
of steel required within the central band. The parameter, β, is equal to the ratio of the
long dimension to the short dimension.
æ 2 ö
As - BW = As - SD ç
÷
è b +1 ø
(9.27)
Reinforced Concrete Substructures
247
The critical section for moment in footings is to be taken at the face of the column.
The critical section for one-way shear is taken at dv from the face of the column. The
critical section for two-way shear is taken as a perimeter at dv/2 from the column or
concentrated load. An exception occurs for any loading conditions which place the
top of the footing in tension at the column interface. The critical shear for such a
condition is at the face of the column.
One-way shear resistance for footings may often be computed using Equation 9.7
with the shear coefficient β = 2. Refer to the AASHTO LRFD BDS Section 5.7.3.4.1
for cases where this may not be acceptable.
Two-way shear resistance for footings constructed with normal-weight concrete is
given in the AASHTO LRFD BDS in Section 5.12.8.6.3 and here in Equation 9.28.
The perimeter, bo, is taken to be dv/2 from the concentrated load. The ratio of the
long side to the short side of the concentrated load area gives the parameter βc.
æ
0.126 ö
Vn = ç 0.063 +
÷ fc¢bo dv £ 0.126 fc¢bo dv
bc ø
è
(9.28)
If Vu is greater than or equal to ϕVn, then either (a) the footing thickness must be
increased, (b) the concrete strength must be increased, or (c) shear reinforcement
must be added. Shear resistance from shear reinforcement in footings is to be computed by Equation 9.8 with θ = 45 degrees.
Section 10.6.5 of the LRFD BDS requires that a trapezoidal stress distribution
beneath footings be used for structural design.
Eccentricity limits for footings with axial load combined with moment are specified for both the Strength and Extreme Event limit states and are as follows:
• Strength limit state eccentricity ≤ 1/3 of the footing dimension (soil)
• Strength limit state eccentricity ≤ 0.45 times the footing dimension (rock)
• Extreme Event limit state eccentricity ≤ 1/3 of the footing dimension for
γEQ = 0.00 (load factor on live load at the Extreme Event limit state)
• Extreme Event limit state eccentricity ≤ 0.40 times the footing dimension
for γEQ = 1.00 (load factor on live load at the Extreme Event limit state)
For seismic design in accordance with the AASHTO LRFD GS, footing dimensions
must satisfy (L−Dc)/(2Hf ) ≤ 2.5. Otherwise, the footing may not be classified as rigid
and special analyses are required. The footing length in the direction of loading is L,
the column diameter of depth in the detection of loading is Dc, and the footing depth
is Hf. Joint shear in footings must also be assessed for bridges in Seismic Design
Categories C and D in accordance with Section 6.4.5 of the LRFD GS.
9.4 PILE CAP DESIGN
Pile cap design requires determination of cap reinforcement and determination of
the maximum and minimum pile loads at the Strength and Service limit states.
Figure 9.1 depicts an example of pile cap and pile configuration. The vertical axial
248
FIGURE 9.1
LRFD Bridge Design
Pile cap example.
load in any pile, pi, may be calculated using Equation 9.29. Pu is the total vertical
load (including self-weight of the pile cap with appropriate load factor) applied at
the center of the pile cap (indicated by the dotted line, representing a column, in the
figure). Mux is the moment about the x-axis and Muy is the moment about the y-axis.
The maximum pile load occurs at the corner for which the second and third terms in
Equation 9.29 take on the positive sign. The minimum pile load occurs at the corner
for which the second and third terms take on the negative sign. With compression
indicated by a positive pile load, negative results indicate tension (uplift) on the pile
in question.
pi =
M ×x
Pu Mux × yi
±
± uyN i
N
2
N
x2
y
å
i =1
i
å
i =1
(9.29)
i
Although some portion of the applied vertical load is likely carried by the soil
underneath the pile cap, the LRFD BDS, in Section 5.12.9, requires that all loads
resisted by the pile cap and its self-weight are to be assumed to be transmitted to
the piles.
Piling may be steel H-piles, precast concrete piles, or steel pipe piles.
Reinforced Concrete Substructures
249
Section 4 of the AASHTO LRFD BCS contains numerous requirements
which need to be considered in addition to design requirements in the AASHTO
LRFD BDS.
Open-end pipe piles may be filled with concrete. ASTM A252 Grade 2 is used for
pipe piles. Although ASTM A252 is common, structures with seismic requirements
may require additional qualification. One example of this from the AASHTO LRFD
BCS is as follows:
“Pipe shall be ASTM A252, but dimensional tolerance as per API 5L and elongation of
25 percent minimum in 2.0 in. The carbon equivalency shall not exceed 0.05 percent”.
API 5L could be specified, but it requires hydrostatic testing and 48.0 in. outside
diameter is the largest diameter covered by API 5L.
The AASHTO LRFD BCS requires that driving stresses for steel piles be limited
to 90% of yield. One rule-of-thumb means for satisfying this is to limit required
Service limit state pile loads to 25% of yield, though, strictly speaking, driving
stresses are to be determined by wave equation analysis. Driving stresses for concrete piles are limited to 0.85f’c minus the effective prestress, if prestressed piles are
used.
Predrilling of a hole with a continuous flight auger or a wet rotary bit may be
required to install piling. Predrilling may be beneficial when driving the pile will
displace the upper soil enough to push adjoining piles out of the proper position, or
to limit vibration in the upper layers. Predrilled holes are typically smaller than the
diameter or diagonal of the pile cross-section and sufficient to allow penetration of
the pile to the required elevation. If subsurface obstructions are encountered, then
the diameter of the predrilled hole may be increased to facilitate pile installation
or to avoid obstructions. Jetting is another method which can be used to facilitate
driving.
Ends of closed-end pipe piles must be closed with a flat plate or a forged or cast
steel conical point, or other end-closure of approved design. End plates must be at
least ¾ inches thick, cut flush with the outer pile wall. The end of the pipe must be
beveled prior to welding to the end plate with a partial penetration groove weld.
Pipe piles are sometimes filled with concrete, depending on the design strategy
employed. Before concrete is placed in a pipe pile, the pile must be inspected to
confirm the full pile length and a dry bottom condition. Any water accumulations in
the pipe piles is to be removed before the concrete is placed. The minimum compressive strength of concrete used for concrete-filled pipe piles is 2.5 ksi or that required
by design, whichever is larger. A slump of not less than 6 inches and not more than
10 inches must be used for the concrete. Concrete must be placed in each pile in a
continuous operation. No concrete is to be placed in a pile until all driving within
a radius of 15 feet of the pile has been completed, or all driving within the 15-foot
radius must be discontinued until the concrete in the last pile cast has set for at least
two days.
In accordance with Section 5.12.9 of the AASHTO LRFD BDS, piles are to be
embedded at least 12 inches into the pile cap, as specified in Section 10.7.1.2. For
250
LRFD Bridge Design
concrete piles, anchorage reinforcement must be provided by either (a) an extension
of the pile reinforcement or (b) the use of dowels. Uplift forces or stresses induced
by flexure shall be resisted by the reinforcement. The steel ratio for the anchorage
reinforcement must be no less than 0.005, and the number of bars must be at least
four. While Section 5.12.9 requires that the anchorage reinforcement be developed
sufficiently to resist a force of 1.25 f yAs, arguments have been made for omitting
anchorage reinforcement, even when uplift is indicated, and eliminating the tension
pile from subsequent load distribution calculations for the load case considered (i.e.,
re-compute the centroid of the pile group, ignoring the tension piles). This does seem
to be a reasonable design approach.
Pile driving equations available for estimating pile resistance from driving
logs include the Gates method and the Engineering News Record (ENR) method.
Resistance factors are ϕ = 0.40 for the Gates method and ϕ = 0.10 for the ENR
method. Equation 9.30 is the Gates formula. Equation 9.31 is the ENR formula. The
resulting nominal load, Rndr, is in kips.
Rndr = 1.75 Ed log10 (10 N b ) - 100
Rndr =
12 Ed
s + 0.1
(9.30)
(9.31)
Although both the Gates and ENR methods are presented in the LRFD BDS, the
units vary between the methods.
• For the Gates formula, Ed must be expressed in units of ft-lbs.
• For the ENR formula, Ed must be expressed in units of ft-kips.
• In the Gates formula, Nb is the number of blows per inch of pile set. Recall
that geotechnical testing typically reports blow counts in blows per foot.
• In the ENR formula, s = 1/Nb, is the pile set in inches per blow.
For additional pile requirements, see Sections 9.6 and 9.9 of this text.
9.5 DRILLED SHAFT DESIGN
A partial set of AASHTO LRFD BDS requirements for geotechnical design of
drilled shafts is summarized below in Equations 9.32 and 9.33 for nominal tip and
side resistance, respectively. With intact rock at least two shaft diameters below the
shaft tip under normal conditions, full tip resistance (qp) and side resistance (qs) for
shafts socketed into rock are given by the following equations.
q p = 2.5qu
(9.32)
qu
pa
(9.33)
qs = pa
Reinforced Concrete Substructures
251
qu is the uniaxial compressive strength of the rock in ksf.
pa is atmospheric pressure, taken as 2.12 ksf.
Resistance factors at the Strength limit state for static analysis of drilled shaft
design geotechnical resistance vary from 0.40 for tip resistance in clay up to as
high as 0.55 for side resistance in sand or rock. The reader is referred to Chapter
10 of the AASHTO LRFD BDS for a complete discussion of applicable resistance
factors.
The geotechnical engineer may provide permissible percentages of tip and side
resistance in combination to determine total geotechnical resistance. For example,
the engineer may determine that 50% of the tip resistance may be directly combined
with 100% of the side resistance (or vice versa, or any other combination of percentages) to establish the total design geotechnical resistance.
The top portion of a drilled shaft socket is sometimes ignored in side-resistance
computations, again as directed by the geotechnical engineer.
9.6 PILE BENT DESIGN
Pile bents are often an economical option for bridges. Pile bents make use of piles
extended to the pier cap and embedded therein, rather than completely in-ground
piles. Figure 9.2 depicts the elements of a typical pile bent. Span lengths are limited
FIGURE 9.2 Typical pile bent.
252
LRFD Bridge Design
by the ability of local constructors to provide, drive, and load-test the piling. The
larger the pile type and driving capacity, the longer will be the span for which a pile
bent bridge proves economical.
For Strength and Service limit states, pile bent design requires flexure-axial
load interaction criteria. Unlike completely in-ground piles, which are often
designed as purely axial elements, the piles in a pile bent are subjected to significant flexure.
The piles in a pile bent may be precast, reinforced concrete, precast, prestressed
concrete, steel HP sections, or steel pipe piles. When pile lengths longer than about
75 ft are expected, it may be advisable to specify pipe piles and to provide splicing
details to develop the full resistance of the pile. Certainly, it is necessary to ensure
that splice locations are far away from anticipated plastic hinge locations for seismic
design.
While hollow, un-filled pipe piles have been used successfully, it may be preferable to use concrete-filled steel tubes when pipe piles in regions of high seismic
hazard are encountered.
Section 9.9 in this text presents a detailed treatment of pipe pile and concretefilled steel tube (CFST) design requirements.
An estimate of the depth to point of fixity is required for the analysis and design
of pile bents. Section 10.7.3.13.4 of the AASHTO LRFD BDS presents, in the commentary, estimates of depth to fixity, in feet, for both clays (Equation 9.34) and sands
(Equation 9.35).
•
•
•
•
•
éE I ù
D f = 1.4 ê p w ú
ë Es û
0.25
éE I ù
D f = 1.8 ê p w ú
ë nh û
0.20
(9.34)
(9.35)
Ep = modulus of elasticity of the pile, ksi
Iw = weak axis moment of inertia for pile, ft4
Es = soil modulus for clays = 0.465 Su, ksi
Su = undrained shear strength of clays, ksf
nh = rate of increase of soil modulus with depth, ksi/ft
Representative ranges of values for nh, based on broad classification, are as follows:
•
•
•
•
•
•
loose sand, dry or moist, nh = 0.417 ksi/ft
loose sand, submerged, nh = 0.208 ksi/ft
medium sand, dry or moist, nh = 1.110 ksi/ft
medium sand, submerged, nh = 0.556 ksi/ft
dense sand, dry or moist, nh = 2.780 ksi/ft
dense sand, submerged, nh = 1.390 ksi/ft
Reinforced Concrete Substructures
253
For clays, Es ranges, again based on broad categorization, are as follows:
• medium stiff clay, Es = 0.347 to 2.08 ksi
• stiff clay, Es = 2.08 to 6.94 ksi
• very stiff clay, Es = 6.94 to 13.89 ksi
Undrained shear strength, Su, generally falls within the following ranges:
• soft soil, Su < 1 ksf
• stiff soil, Su = 1–2 ksf
• very stiff soil, Su exceeds 2 ksf
Other rough estimates of depth to fixity are given in the literature (Priestley et al.,
1996). For moment, the depth to fixity has been estimated at approximately 1 to 2
diameters. For deflection, the estimate is 4 to 5 diameters.
Section 5.12.9.3 of the LRFD BDS requires that concrete piles, whether reinforced or prestressed, must have a cross-sectional area no less than 140 in2 unless
exposure to saltwater is possible, in which case the minimum cross-sectional area is
220 in2. Concrete used for prestressed piles must have a specified 28-day concrete
strength no less than 5 ksi.
For prestressed concrete piling, the uniform compressive stress on the cross section, after prestress losses, must be no less than 0.700 ksi.
For bridges in LRFD BDS Seismic Zone 1 (LRFD GS Seismic Design Category
A), transverse, spiral reinforcement is required over the entire pile length for prestressed concrete piles, whether round, rectangular, or other. The main spiral reinforcement must be spiral wire not less than W3.9 at a pitch of no more than 6 inches
for piles having the least dimension no greater than 24 inches. For larger piles, the
required transverse spiral is to be no less than W4.0 wire with a pitch not to exceed
4 inches. See Section 5.12.9.4.3 of the LRFD-BDS for spacing limits near the ends
of piles.
For LRFD BDS Seismic Zones 2, 3, and 4 (LRFD GS Seismic Design Categories
B, C, and D), enhanced requirements must be satisfied for piles. Transverse reinforcement must be at least No. 3 bar at a pitch not to exceed 9 inches.
For Seismic Zones 3 and 4, the upper end of the pile must contain confinement
reinforcement as a potential plastic hinge location. The potential hinge region
extends from the bottom of the pile cap for a distance no less than either (a) two pile
diameters or (b) 24 inches.
For pile bents, the top potential hinge region extends from the bottom of the cap
for a distance no less than the greater of (a) the maximum cross-section dimension, (b) one-sixth of the clear height, or (c) 18 inches. For in-ground hinges in pile
bents, the bottom potential hinge region extends from three pile diameters below the
point of maximum, in-ground moment to one diameter above the maximum moment
point. However, the top limit for the in-ground potential hinge region shall not be
less than 18 inches above the top of the ground.
254
LRFD Bridge Design
For circular columns and piles, Equation 9.36 provides the required volumetric
spiral or seismic hoop reinforcement for plastic hinge regions. The core diameter,
dc, is measured to the outside diameter of the spiral or hoop. The pitch, s, is measured vertically to the center of the spiral or hoops. Asp is the cross-sectional area of
the spiral or hoop bar. The specified minimum yield strength, f yh, is that for the spiral or hoop bar, not necessarily the same as that for longitudinal bars. The specified
concrete strength, f’c, is used for the calculations, rather than the expected concrete
strength, f’ce. Certainly, Equation 9.37 from Section 5.6.4.6 of the LRFD BDS, for
non-seismic confinement requirements, should always be checked as well. The
core area, Ac, is based on the diameter to the outside of the spiral or hoop as well.
Equations 9.38 and 9.39 provide the required confinement reinforcement, Ash,
within a spacing equal to s, for rectangular piles and columns. The dimension, hc,
is measured to the outside of the transverse bars in the direction of loading. The
spacing, s, must exceed neither (a) 4 inches, nor (b) one-quarter of the least crosssectional dimension.
4 Asp
f¢
³ 0.12 c
dc s
f yh
(9.36)
4 Asp
æA
ö f¢
³ 0.45 ç g - 1 ÷ c
dc s
A
è c
ø f yh
(9.37)
rs =
rs =
Ash ³ 0.12shc
fc¢
f yh
æA
ö f¢
Ash ³ 0.30shc ç g - 1 ÷ c
è Ac
ø f yh
(9.38)
(9.39)
For either round or rectangular columns and piles, the yield strength of the transverse reinforcement, f yh, is the minimum specified value for the reinforcement used,
but must not exceed 75 ksi.
For pile bent design by the displacement-based provisions of the LRFD GS, the displacement ductility demand, μD, is not to exceed 4. For single column bents, μD is limited to 5, and for multi-columns bents with above-ground hinging, μD is limited to 6.
For slenderness effects, it is often necessary to determine an appropriate effective
length factor, K, from section 5.6.2.5 of the LRFD BDS. When integral abutments
are used at both ends of the bridge (i.e., no expansion joints), it may be possible to
treat pile bents as braced out-of-plane (K = 1.0) and as rigid frames in-plane (K = 1.2).
Provided sufficient stiffness is available in the longitudinal direction from the integral abutments, the assumption may or may not be valid. With expansion abutments,
it would most likely seem advisable to treat the pile bent as unbraced out-of-plane
(K = 2.0).
Reinforced Concrete Substructures
255
9.7 BRIDGE PIER DISPLACEMENT CAPACITY
UNDER SEISMIC LOADING
Bridge design for seismic effects has increasingly become displacement-based, as
opposed to the original, decades-old, force-based provisions. Pushover analyses by
computer modeling and approximate hand calculations are both useful.
Approximate pushover analysis of concrete piers calculated by hand may be
accomplished using an analysis incorporating the Mander model for confined concrete along with procedures outlined in the literature (Priestley et al., 2007) and
reproduced here in Equations 9.40 through 9.48. Displacement estimates may be
done by computer modeling or by approximate hand-calculation-based equations,
such as those presented here in Equations 9.49 through 9.56.
ì
æey ö
ï 2.25 ç D ÷ , circular columns
è
ø
ï
fy @ í
ï2.10 æ e y ö , rectangular coolumns
ç D÷
ïî
è
ø
(9.40)
c
P
@ 0.20 + 0.65
D
fce¢ Ag
(9.41)
1.4 r v f yhe su
ì
, damage control limit state
ï0.004 +
fcc¢
e cu = í
ï
0.004, serv
viceability limit state
î
(9.42)
ì0.06 to 0.09, damage control limit state
e su = í
0.015, serviceability limit state
î
(9.43)
rcc =
As
Area of longitudinal reinforcement
=
Ac Area of core enclosed by centerlines of hoop or spiral
(9.44)
ì Aspp ds 4 Asp
ï p 2 = d s ® circular hoops or spirals
s
ï ds s
rv = í 4
ï
A
Asx
+ sy ® rectangular hoops
ï r vx + r vy =
shcy shcx
î
(9.45)
æ
f¢ ö
7.94 fl¢
fcc¢ = fco¢ ç -1.254 + 2.254 1 +
-2 l ÷
ç
fco¢
fco¢ ÷ø
è
(9.46)
256
LRFD Bridge Design
ì
ï
ï
ï
ï
ï
ï
ke = í
ï
ïæ
ï ç1 ïç
ïè
ï
î
æ
s¢
ç 1 - 2d
s
è
1 - rcc
s¢
12 ds
1 - rcc
( wi¢ )
2
ö
÷
ø ® circular hoop effectiveness coefficient
® circular spiral effectiveness coefficient
öæ
s¢ ö æ
s¢ ö
å 6b d ÷÷ø çè1 - 2b ÷ø çè1 - 2d ÷ø
c c
c
c
1 - rcc
® rectangulaar hoop coefficient
1
fl¢ = ke r v f yh ® lateral confining stress on concrete
2
(9.48)
2
1
D y = fy ( Lc + LSP )
3
(9.49)
L ö
æ
D P = (fu - fy ) LP ç Lc + LSP - P ÷
2 ø
è
(9.50)
æ f
ö
k = 0.20 ç u - 1 ÷ £ 0.08
è fy
ø
(9.51)
LP = kLc + LSP ³ 2 LSP
(9.52)
LSP = 0.15 f ye dbl
(9.53)
Du = D y + D P
(9.54)
m=
Du
Dy
ì e cu
ïï c
fu = Min í
ï e su
ïî d - c
•
•
•
•
•
(9.47)
(9.55)
(9.56)
Lc is the distance from the critical section to the point of contra-flexure
LSP is the strain penetration distance
LP is the plastic hinge length
d is the column depth
c is the distance from the compression face of the column to the neutral axis
257
Reinforced Concrete Substructures
•
•
•
•
•
•
•
•
•
•
•
•
•
•
ds is the hoop or spiral diameter measured to the center of the hoop or spiral
s is the hoop or spiral pitch measure to the center of the spiral or hoop
s’ is the clear hoop or spiral pitch
Asp is the area of the spiral bar
f’co is the specified concrete strength at 28 days
bc is the out-to-out width measured to the center of the rectangular hoop
dc is the out-to-out height measured to the center of the rectangular hoop
w’ is the clear spacing between adjacent longitudinal bars
εcu is the ultimate concrete compressive strain for a particular limit state
εsu is the ultimate steel tensile strain for a particular limit state
ϕy is the section yield curvature
ϕu is the section ultimate curvature
Δy is the lateral yield displacement occurring over the length, LC
ΔP is the lateral plastic displacement occurring over the length, LC
For rigid frame behavior with plastic hinging at the top and bottom of the column, LC
is one-half of the column height and displacements (Δy and ΔP) are twice that given
by the equations above. This does not apply to the implicit displacement equations
given below, as the rigid frame versus cantilever behavior is incorporated into the
factor, Λ, in the implicit equations.
For Seismic Design Categories B and C, approximate displacement capacity equations are provided in the AASHTO LRFD GS (AASHTO, 2011). The expressions
are presented here in Equations 9.57, 9.58, and 9.59. For Seismic Design Category
B, Equation 9.57 is applicable. For Seismic Design Category C, Equation 9.58 is
applicable.
(
)
(9.57)
(
)
(9.58)
DCL = 0.12 H o -1.27 ln ( x ) - 0.32 ³ 0.12 H o
DCL = 0.12 H o -2.32 ln ( x ) - 1.22 ³ 0.12 H o
x=
LBo
Ho
(9.59)
• Λ = 1 for fixed–free column end conditions (cantilever)
• Λ = 2 for fixed–fixed column end conditions (rigid frame)
• Bo is the column diameter for circular columns, or the dimension parallel to
the direction in which displacement capacity is being calculated, for rectangular columns, feet
• Ho is the clear column height, feet
Note that Ho and Bo are in feet, whereas the displacement capacity from the equations is in inches.
The displacement capacity equations are intended for use with reinforced concrete column substructures with a minimum clear height of 15 feet. Attempts to use
the equations for other situations will likely result in serious error.
258
LRFD Bridge Design
9.8 THE ALASKA PILE BENT DESIGN STRATEGY
One innovative design concept for pile bent bridges has been developed by the Alaska
Department of Transportation and Public Facilities based on research conducted at
the University of California at San Diego (Silva and Seible, 1999).
The method relies on a reinforced concrete section at the cap soffit, with the
steel tube providing only external confinement to the reinforced concrete. For the
in-ground hinge, the cross section consists of the filled tube. Proof of the method
has been provided in research (Silva and Sritharan, 2011). The strategy is particularly effective in minimizing damage to the cast-in-place cap. A two-inch gap is
required between the top of the steel tube and the cap soffit. The Washington State
Department of Transportation (WSDOT, 2020) has also recognized this alternative
design strategy.
Should such a strategy be adopted for a particular project, then the literature
should be referenced, particularly with regard to required joint reinforcement.
9.9 CONCRETE FILLED STEEL TUBES (CFST)
The design of concrete-filled steel tubes (CFST) is covered in both the AASHTO
LRFD BDS and the AASHTO LRFD GS.
Section 6.6.6.1 of the AASHTO LRFD BDS requires that CFST members expected
to experience plastic hinging should be designed according to the AASHTO LRFD
GS. However, Section 7.6 of the AASHTO LRFD GS requires that CFST expected
to experience plastic hinging be designed according to the AASHTO LRFD BDS,
Sections 6.9.2.2, 6.9.5, and 6.12.3.2.2, in addition to the GS requirements.
CFST are used for piles, drilled shafts, columns, and other members subject to
axial compression or axial compression and flexure. In the AASHTO LRFD BDS,
the design of composite CFST may be performed in accordance with either:
• Section 6.9.6 along with Section 6.12.2.3.3 for composite CFST design, or
• Section 6.9.5 along with Section 6.12.2.3.2 for partially composite CFST
design
The provisions specified in LRFD-BDS Section 6.9.6 incorporate a great wealth of
research performed since the development of Section 6.9.5 and tend to reduce uncertainty and increase the accuracy of analytical results. Nonetheless, it appears that
AASHTO recommends GS Section 7.6 in combination with BDS Sections 6.9.2.2,
6.9.5, and 6.12.3.2.2 for CFST design with plastic hinging.
For Seismic Design Categories C and D, ductile concrete-filled steel pipe, as
defined in Section 7.6 shall be made of steels satisfying the requirements of either
(a) ASTM A 53 Grade B or (b) API 5L X52. For ASTM A 53 Grade B tubes, the
expected yield stress, Fye, is to be 1.5 times the nominal yield stress for overstrength
calculations. For API 5L X52 tubes, the expected yield stress is 1.2 times the nominal yield stress.
Overstrength moments are to be determined using expected yield stress for the
tube, expected concrete strength (1.3f’c), and an overstrength factor, λmo, applied to
259
Reinforced Concrete Substructures
the calculated moment. The overstrength factor is 1.2 for structural steel, 1.2 for reinforced concrete using A706 reinforcing bars, and 1.4 for reinforced concrete using
A615 reinforcing bars.
9.9.1 CFST Design in Accordance with BDS
Sections 6.9.6 and 6.12.2.3.3
Requirements on tube dimensions and concrete fill given in Equations 9.60 and 9.61
are applicable when Section 6.9.6 of AASHTO LRFD BDS is adopted for CFST
design. The outside tube diameter is D, the tube thickness is t, E is Young’s modulus
(for steel tubes, E = 29,000 ksi), Fyst is the specified minimum yield strength of the
tube material.
D
E
£ 0.15
t
Fyst
(9.60)
fc¢ ³ Max {3.0 ksi, 0.075Fyst }
(9.61)
The resistance factor, ϕc, for combined axial compression and flexure, is 0.90 for
CFST design. A nominal P–M interaction diagram, reduced by ϕc, is used to assess
the adequacy of the CFST subjected to applied Strength limit state loads, Pu and Mu.
Development of the nominal P–M–interaction curve requires initial determination
of axial resistance, ϕPn. The axial resistance is determined using Equations 9.62
through 9.66. The area of concrete fill is Ac. The cross-sectional area of the tube is
Ast, with corresponding yield strength Fyst. The area of internal reinforcing bars is
Asb, with corresponding yield strength, Fyb. The un-factored axial load is P.
Po = 0.95 fc¢Ac + Fyst Ast + Fyb Asb
(9.62)
EI eff = EI st + EI si + C ¢Ec I c
(9.63)
C ¢ = 0.15 +
P
Ast + Asb
£ 0.90
+
Po Ast + Asb + Ac
Pe =
p 2 EI eff
( KL )
2
P
ìé
( Po / Pe ) ù
P , if e > 0.44
ïï ë0.658
û o
Po
Pn = í
P
ï
0.877Pe , if e £ 0.44
ïî
Po
(9.64)
(9.65)
(9.66)
Material-based P–M interaction may be established using either (a) the plastic stress
distribution method (PSDM) or (b) the strain compatibility method (SCM). The
260
LRFD Bridge Design
PSDM is detailed in AASHTO LRFD BDS, Section 6.12.2.3.3. The interested reader
is directed to this Section of the AASHTO LRFD BDS for further guidance.
Shear resistance of the concrete-filled steel tubes is taken to be equal to the
shear resistance of the tube alone, according to the AASHTO LRFD BDS, Section
6.12.3.2.2. See Section 9.9.3 of this text as well.
9.9.2 CFST Design by BDS Sections 6.9.5 and
6.12.3.2.2 and GS Section 7.6
Given the difficulty in interpreting BDS Section 6.9.6, combined with the recommendation in the AASHTO LRFD GS that BDS Section 6.9.5, in combination with
GS Section 7.6, be used for CFST design, a more detailed discussion of this method
is presented here.
The wall thickness of CFST designed using BDS 6.9.5 must satisfy Equation 9.67.
The concrete must satisfy Equation 9.68. The specified yield strength of the tube
and any reinforcement is not to exceed 60 ksi. The tube area should equal or exceed
4 percent of the total cross-sectional area, or the member should be designed as a
reinforced concrete column using Section 5 of the LRFD BDS.
D
E
£ 0.11
t
Fyst
(9.67)
3.0 ksi £ fc¢ £ 8.0 ksi
(9.68)
Compressive resistance in Section 6.9.5 is taken to be equal to that given by Equations
9.69 through 9.72. When moment magnification is used to estimate second-order
effects, Equation 9.73 is applicable for the computation of the Euler buckling load.
Flexural resistance is given by Equation 9.74. The plastic moment, Mps, is for the steel
tube alone. The yield moment, Myc, is for the composite section. The modular ratio,
n, = ES/Ec. The radius of gyration, rs, in Equation 9.71 is that of the tube alone.
æA
Fe = Fy + Fyr ç r
è As
ö
æ Ac ö
÷ + 0.85 fc¢ ç A ÷
ø
è sø
é æ 0.40 ö æ Ac ö ù
Ee = E ê1 + ç
÷ ç ÷ú
ë è n ø è As ø û
(9.69)
(9.70)
2
æ KL ö Fe
l =ç
÷
è rsp ø Ee
(
(9.71)
)
ì 0.66l Fe As , if l £ 2.25
ï
Pn = í æ 0.88F A ö
e s
ïç
÷ , if l > 2.25
l
ø
îè
(9.72)
261
Reinforced Concrete Substructures
Pe =
As Fe
l
(9.73)
ì
D
E
M ps , if
<2
ï
t
F
y
ï
Mn = í
ï M , if 2 E £ D £ 8.8 E
ï yc
Fy
t
Fy
î
(9.74)
Requirements from the ASHTO LRFD GS for combined axial compression and flexure are summarized in Equations 9.75 through 9.87. Pn is given by Equation 9.72. Mn
is given by Equation 9.87. Presumably, when Extreme Event limit states are being
analyzed, the resistance factors on Pn and Mn should both be equal to 1.0 in the
equations. This method underestimates flexural resistance by, on average, 25 percent
for D/t ratios up to 0.14E/Fy. Engineers may wish to incorporate this into estimates
of maximum load delivered to elements intended to remain elastic during extreme
events. Equation 9.78 for Pro is simply Equation 9.72 for Pn using an appropriate
resistance factor, along with λ = 0.
Three methods are available for the determination of Mn. A closed-form method
based on exact geometry requires recursive solution of Equation 9.80 for β, the angle
(in radians) subtended by the neutral axis chord to the tube center, and subsequent
solution for additional parameters to determine Mn as given by Equation 9.87.
æ M ö
Pu
+ B ç u ÷ £ 1.0
f Pn
è f Mn ø
(9.75)
Mu
£ 1.0
f Mn
(9.76)
Prc = 0.75 Ac fc¢
(9.77)
Pro = f Fe As
(9.78)
Prc
Pro
(9.79)
B = 1-
é æb ö
æb
As Fy + 0.25D 2 fc¢ êsin ç ÷ - sin 2 ç
2
è
ø
è2
ë
b=
2
0.125D fc¢ + DtFy
Cr = Fy b
Dt
2
ö æb
÷ tan ç 4
ø è
öù
÷ú
øû
(9.80)
(9.81)
262
LRFD Bridge Design
æb ö
bc = D sin ç ÷
è2ø
(9.82)
bc
æb ö
tan ç ÷
2
è4ø
(9.83)
a=
é b D 2 bc
Cr¢ = fc¢ ê
2
ë 8
æD
öù
ç 2 - a ÷ú
è
øû
(9.84)
æ 1
1ö
e = bc ç
+ ÷
è 2p - b b ø
(9.85)
æ 1
ö
bc2
+
e¢ = bc ç
ç 2p - b 1.5b D 2 - 6bc ( 0.5D - a ) ÷÷
è
ø
(9.86)
f M n = f f éëCr e + Cr¢e¢ùû
(9.87)
Another closed-form solution for Mn, based on approximate geometry, is simpler and
given by Equations 9.88 and 9.89. The section modulus, Z, of the tube alone appears
in the equations. This alternative closed-form solution results in lower estimates for
Mn compared to those obtained using Equation 9.87. Hence, the engineer may wish
to adjust expected moments for capacity design further when using this method to
estimate flexural resistance.
é
3
æ2
ö ù
f M n = f f ê Z - 2thn2 Fy + ç ( 0.5D - t ) - ( 0.5D - t ) hn2 ÷ fc¢ú
3
è
ø û
ë
(
)
hn =
Ac fc¢
2 Dfc¢ + 4t ( 2 Fy - fc¢ )
(9.88)
(9.89)
A third method for determining axial compression interaction with flexure is to perform a strain compatibility analysis using appropriate constitutive models for the
various materials.
9.9.3 Steel Tube Design without Concrete Fill
With no concrete fill, steel tube wall slenderness, D/t, is limited to 0.45E/Fy in the
AASHTO LRFD BDS, Section 6.12.2.2.3. Flexural resistance of steel tubes without
fill is determined as the lesser of that for the limit states of yielding and local buckling, given by Equations 9.90 and 9.91, respectively. The elastic, S, and plastic, Z,
section moduli are given in Equations 9.92 and 9.93 for a tube with outer diameter,
D, and inner diameter, Di = D–2t.
263
Reinforced Concrete Substructures
M n = Fy Z
(9.90)
ìæ 0.021E
D 0.31E
ö
ïç D /t + Fy ÷ S, if t £ F
ø
y
ïè
Mn = í
D 0.31E
0.33E
ï
S, if
>
ïî
D /t
t
Fy
(9.91)
S=
(
p D 4 - Di4
Z=
)
32 D
D 3 - Di3
6
(9.92)
(9.93)
The shear resistance of steel tubes without fill is given by Equation 9.94 from Section
6.12.1.2.3 of the AASHTO LRFD BDS. The shear length, Lv, is the distance between
points of zero and maximum shear in the member under question. Should a point of
zero shear not exist, then Lv is taken as the full member length.
Vn = 0.5Fcr Ag
(9.94)
ì 1.60 E
£ 0.58Fy
5
ï
4
ï Lv æ D ö
ïï D çè t ÷ø
Fcr = Max í
ï 0.78 E £ 0.58 Fy
ï æ D ö32
ï ç ÷
ïî è t ø
(9.95)
The resistance factor for shear, ϕv, = 1.00. The resistance factor for axial compression, ϕc, = 0.95, and for axial compression combined with flexure, ϕc, = 0.90.
9.9.4 CFST Design for Extreme Event Limit States
For analysis and design of CFST at the Extreme Event limit state for earthquake, ice,
and collision loading, a model for inelastic behavior is required. The LRFD BDS is
relatively silent on the subject of appropriate models for such analyses.
The Port of Los Angeles (POLA, 2010) and the Port of Long Beach (POLB, 2015)
each provide estimated strain limits for both hollow tube and CFST used in wharves.
The Washington State Department of Transportation (WSDOT, 2020) also provides
useful information for CFST elements.
POLA and POLB each specify three levels of earthquake ground motion for the
design of wharves, with corresponding strain limits for each level. The three levels
of ground motion correspond to:
264
LRFD Bridge Design
• Operating Level Earthquake (OLE) – 50% probability of exceedance in 50
years (72-year mean recurrence interval)
• Contingency Level Earthquake (CLE) – 10% probability of exceedance in
50 years (475-year mean recurrence interval)
• Design Earthquake (DE) – 2% probability of exceedance in 50 years (2,475year mean recurrence interval). The MRI (mean recurrence interval) for the
DE is not explicitly stated, but reference is made to ASCE 7.
Strain limits for both POLA and POLB are summarized in Table 9.2. In-ground
plastic hinge strain limits are based on depth to fixity of no more than ten pile diameters. Strain limits for pipe piles at the pile top are based on the use of a concrete plug
with dowels into the cap. However, these strain limits do not account for the possibility of local buckling of the tube wall and are likely to only be valid for extremely low
D/t values. Equation 9.96, from recent research (Harn et al., 2019), provides strain
criteria which capture potential local buckling and are recommended over Table 9.2
values for DBE (design basis earthquake) evaluations.
æDö
e s £ 10 ç ÷
è t ø
-2
(9.96)
Parameters in Table 9.2 are defined as follows.
•
•
•
•
•
•
εc = extreme compressive strain in concrete
εsd = extreme tensile strain in dowel reinforcement
εs = steel shell extreme fiber strain
εsmd = strain at peak stress for dowel reinforcement
εp = extreme tensile strain in prestressing strand
ρs = confining steel volumetric ratio
TABLE 9.2
POLA/POLB Steel Tube Strain Limits
Pile
OLE
CLE
Concrete
(pile top)
εc ≤ 0.005
εsd ≤ 0.015
εc ≤ (0.005+1.1ρs) ≤ 0.025
εsd ≤ 0.6εsmd ≤ 0.060
No limit on εc
εsd ≤ 0.8εsmd ≤ 0.080
DBE
Concrete
(in-ground)
εc ≤ 0.005
εp ≤ 0.015
εc ≤ (0.005+1.1ρs) ≤ 0.008
εp ≤ 0.025
εc ≤ (0.005+1.1ρs) ≤ 0.025
εp ≤ 0.035
Pipe pile
(pile top)
εc ≤ 0.010
εsd ≤ 0.015
εc ≤ 0.025
εsd ≤ 0.6εsmd ≤ 0.060
No limit on εc
εsd ≤ 0.8εsmd ≤ 0.080
Pipe pile
(in-ground)
εs ≤ 0.010
εs ≤ 0.025
εs ≤ 0.035
CFST
(in-ground)
εs ≤ 0.010
εs ≤ 0.035
εs ≤ 0.050
Reinforced Concrete Substructures
265
The WSDOT guidance permits slightly relaxed criteria, compared to LRFD BDS
criteria. The slenderness limit for local buckling effects is given by Equation 9.97
for elastic elements or Equation 9.98 for plastic, ductile elements. WSDOT also permits the calculation of shear resistance to include the contribution of concrete fill as
shown in Equation 9.99. The coefficient, g4, is equal to 1.0 under the assumption of
fully composite behavior. A reduced value may be required if such conditions are
not present.
D
E
£ 0.22
t
Fyst
(9.97)
D
E
£ 0.15
t
Fyst
(9.98)
fVn = j ( g4 ) é0.6 f y As + 0.0316 ( 3 ) Ac fc¢ ù
ë
û
(9.99)
9.10 TWO-WAY SHEAR
In addition to requirements for one-way (beam) shear in reinforced concrete, pile caps
and similar elements must be assessed for two-way (formerly referred to as “punching”) shear. Such checks are typically made at the column and at the most heavily
loaded pile. Equation 9.100 defines the design shear resistance for two-way behavior in elements without transverse reinforcement and comprised of normal-weight
concrete. The parameter, βc, is the ratio of the long side to the short side of the area
through which the load is transmitted. The perimeter, bo, surrounds the loaded area at
dv/2 from each edge of the loaded area. Recall that the shear depth, dv, is the distance
between the resultant cross-sectional compressive and tensile forces, but must not be
less than either 0.72h or 0.90de. Since the two-way resistance is that of the pile cap,
then dv is the variable for the pile cap (not the loaded area (column or pile)).
æ
0.126 ö
fVn = 0.90 ç 0.063 +
÷ fc¢ ( bo dv ) £ 0.126 fc¢ ( bo dv )
bc ø
è
(9.100)
Refer to the AASHTO LRFD BDS, Section 5.12.8.6.3, for a full discussion of twoway shear in reinforced concrete pile caps.
9.11 FATIGUE RELATED ISSUES IN REINFORCED CONCRETE
The AASHTO LRFD BDS explicitly exempts concrete deck slabs from fatigue
investigation in Section 5.5.3. This same section does require fatigue investigation
for reinforcement bar stress in areas where total unfactored dead load (DC plus
DW) compressive stress is less than 1.75 times the Fatigue limit state stress range
(Fatigue I, infinite life, with impact of 15% included). Equation 9.101 gives the basic
fatigue requirement. Equation 9.102 provides the criteria to be used for straight
266
LRFD Bridge Design
reinforcing bars as well as for welded wire reinforcement, both without welding in
the high-stress region, defined as one-third of the span on each side of the point of
maximum moment. For locations with bar welds, Equation 9.103 applies. For regular
lap-spliced bars, the fatigue threshold, (ΔF)TH is taken to be equal to 4.0 ksi.
In the equation for fatigue criteria for reinforcement, the yield stress, f y, is not to be
less than 60 ksi, nor more than 100 ksi, regardless of actual, specified yield strength.
The bar stress, f min, is equal to the minimum live load stress from the Fatigue I
limit state combined with either (a) the unfactored DC and DW load effects or (b) the
unfactored permanent loads, shrinkage, and creep-induced effects. The stress, f min,
is positive if tension, negative if compression.
Section properties for fatigue calculations are to be based on cracked sections
whenever the sum of stresses, due to unfactored permanent loads combined with the
Fatigue I limit state loading, is tensile and exceeds 0.095(f’c)1/2.
g ( Df ) £ ( DF )TH
(9.101)
fmin
fy
(9.102)
( DF )TH = 18 - 0.36 fmin
(9.103)
( DF )TH = 26 - 22
9.12 ABUTMENT DESIGN
For stub abutments integral with the concrete deck, the design of abutments is relatively simple. With integral abutments, the superstructure effectively prevents large
bending moments in the backwall. However, if the construction sequence requires
backfilling prior to superstructure installation, the backwall will still need to be
designed to resist the fill as a retaining wall.
The abutment beam of a stub abutment on piles may be designed as a continuous, reinforced concrete beam spanning between piles. It is seldom worth the extra
effort of a computer model for this design, and bounding assumptions may be made.
One such assumption is the plunging of a single pile, producing a span length for the
continuous abutment beam equal to twice the pile spacing. While it is certainly possible to place superstructure girder reactions at constructed locations to determine
shears and moments in the continuous abutment beam, it is likely to result in an economical, reasonable solution by assuming worst-case locations of girder reactions
for shear and moment.
Initially, the total superstructure reaction at the Service and Strength limit states
must be determined and combined with self-weight of the abutment backwall and the
abutment beam. One proven method for determining the number of point-bearing
piles required to support the loads is to assume an equal distribution of vertical
load to the piles, and to specify a number of piles such that the Service limit state
axial load produces a uniform compressive pile stress no more than 25% of the yield
strength of the pile. For friction piles, the Gates formula or the ENR formula may be
used to estimate pile resistance from driving logs. Such piles are typically installed
Reinforced Concrete Substructures
267
with driving logs obtained and compared with results from load testing of a certain
percentage of the production piles. The load test permits the use of a higher resistance factor in determining pile resistance at the Strength limit state.
For non-stub abutments, the design is more complex. For large fill behind abutment walls, it becomes necessary to design a bona fide retaining wall system for the
construction phase, with final bridge loads also accommodated.
A cross-section for a typical stub-type, integral abutment is shown in Figure 9.3. A
more complex, non-stub type, fixed abutment is shown in cross section in Figure 9.4.
FIGURE 9.3
Cross section of a typical stub-type integral abutment.
268
FIGURE 9.4 Non-stub fixed abutment cross section.
LRFD Bridge Design
Reinforced Concrete Substructures
269
9.13 SOLVED PROBLEMS
Problem 9.1
A three-column bent is shown in Figure P9.1. Concrete strengths are,
f’c = 4 ksi and f’ce = 5.2 ksi. Steel yield stress values are f y = 60 ksi and
f ye = 68 ksi.
Columns are 48 inches in diameter with 30 #9 longitudinal bars and #5
hoops at 5 inches on centers. A 706 reinforcement with reduced ultimate
strain εsu = 0.09 is used for the hoops and the longitudinal bars. Columns are
designed for full fixity at both top and bottom of the column.
Girder reactions for the conditions under consideration include onehalf of the live load and are 443 kips for each of the exterior girders, and
509 kips for each of the interior girders.
The total cap length is 45 feet.
FIGURE P9.1
Problem P9.1.
270
LRFD Bridge Design
Determine the displacement capacity of the bent using:
a) AASHTO LRFD GS equations for implicit displacement capacity for
Seismic Design Category B.
b) AASHTO LRFD GS equations for implicit displacement capacity for
Seismic Design Category C.
c) A simplified pushover analysis, given that the plastic shear, VP, for the
bent has been determined to be 993 kips.
Problem 9.2
A pier cap is shown in Figure P9.2. f’c = 4 ksi and f y = 60 ksi. Assess the
adequacy of the proposed cross section for two cases:
a) Positive moment (tension in the bottom of the cap)
Mu = 3,720 ft-k, Vu = 56 kips (Strength I limit state)
Mu = 1,950 ft-k, Vu = 30 kips (Service I limit state)
b) Negative moment (tension in the top of the cap)
Mu = -2,700 ft-k, Vu = 180 kips (Strength I limit state)
Mu = -1,680 ft-k, Vu = 112 kips (Service I limit state)
Use the AASHTO equations and hand calculations. Verify the results using
Response 2000.
Problem 9.3
The pile bent shown in Figure P9.3 is used as an intermediate pier in a shortspan continuous bridge. The CFST piles are 20 inch in diameter with 5/8 inch
in wall thickness and fabricated in accordance with ASTM A53 Grade B,
FIGURE P9.2
Problem P9.2.
Reinforced Concrete Substructures
FIGURE P9.3
271
Problem P9.3.
Fy = 35 ksi. Concrete core has 28-day compressive strength, f’c = 3 ksi. For
the conditions shown, each of the five girder-bearing locations carries a vertical reaction of 117 kips. The core is reinforced with six #8 bars.
The cap is 4 ft by 4 ft in cross section. The lateral load shown represents
an incremental load to be applied at the superstructure center of gravity in
a pushover analysis.
Estimate the lateral load resistance of the bent using one of the methods
in AASHTO for the calculation of CFST flexural resistance.
Problem 9.4
For the pier column shown in Figure P9.4, the actions at the column base at
the Strength limit state are:
Pu = 3,770 kips
VuT = 96 kips
MuL = 13,234 ft-kips
VuL = 67 kips
MuT = 5,915 ft-kips
The unsupported column height is 67 feet. The column is 6 ft in diameter
with 44 #10 bars. Concrete f’c is equal to 3 ksi. Grade 75 longitudinal bars
are used. Grade 60, #5 hoops at 18 inches are used for the full height. Clear
cover is 2 inches. Determine the required resistance, Mu, and the design
resistance, ϕMn. Include slenderness effects.
Problem 9.5
For the concrete girder option of the Project Bridge, the pier cap shear
and moment for one particular Strength limit state load combination are
272
FIGURE P9.4
LRFD Bridge Design
Problem P9.4
provided below. The cap cross section is shown in Figure P9.5. Dimensions
shown are inches. Measured from the top surface, the top four bar layers are
at 4 inches, 8 inches, 17 inches, and 26 inches. The specified 28-day concrete compressive strength is f’c = 4 ksi. Steel bar yield stress, f y = 60 ksi. The
transverse reinforcement shown is equivalent to four legs of #5 closed stirrups spaced at 6 inches apart. Determine the design shear resistance, ϕVn,
and flexural resistance, ϕMn, at the section under consideration. Compare
these resistance values with the required resistance values.
Mu = −1,462 ft-kips (tension in the top of the cap)
Vu = 311 kips
Problem 9.6
For the concrete girder option of the Project Bridge, the column demands
for one particular Strength limit state load combination are given below.
Reinforced Concrete Substructures
FIGURE P9.5
Problem 9.5
FIGURE P9.6
Problem 9.6.
273
The column cross section is shown in Figure P9.6. Dimensions shown are
inches. The diameter of the bar circle is 40 inches. Determine the design
resistance values, ϕMn, ϕVn, and ϕPn, and compare with the required values.
Pu = 1,231 kips
Mux = 1,054 ft-kips
Muz = 345 ft-kips
Vuz = 44 kips
Vux = 30 kips
274
LRFD Bridge Design
FIGURE P9.7
Problem 9.7.
FIGURE P9.8
Problem 9.8.
275
Reinforced Concrete Substructures
Problem 9.7
For the column base loads given in Problem 9.6, determine the maximum
pressure using the spread footing dimensions shown in Figure P9.7. The
footing has 2 feet of soil fill over the top of the footing. Soil unit weight is
120 pounds per cubic foot.
Problem 9.8
For the column base loads given in Problem 9.6, determine the maximum pile
reaction using the pile cap dimensions shown in Figure P9.8. The pile cap has
2 feet of soil fill over the top of the footing. Soil unit weight is 120 pounds per
cubic foot. Determine the required blow count during pile driving using the
Gates formula and the ENR formula with appropriate resistance factors. The
pile hammer weighs 7,930 pounds and has a stroke of 127 inches.
PROBLEM 9.1
TEH
1/5
276
PROBLEM 9.1
LRFD Bridge Design
TEH
2/5
277
Reinforced Concrete Substructures
PROBLEM 9.1
TEH
3/5
278
PROBLEM 9.1
LRFD Bridge Design
TEH
4/5
279
Reinforced Concrete Substructures
PROBLEM 9.1
TEH
5/5
PROBLEM 9.2
TEH
1/5
280
PROBLEM 9.2
LRFD Bridge Design
TEH
2/5
281
Reinforced Concrete Substructures
PROBLEM 9.2
TEH
3/5
282
PROBLEM 9.2
LRFD Bridge Design
TEH
4/5
283
Reinforced Concrete Substructures
PROBLEM 9.2
TEH
5/5
284
PROBLEM 9.3
LRFD Bridge Design
TEH
1/5
285
Reinforced Concrete Substructures
PROBLEM 9.3
TEH
2/5
286
PROBLEM 9.3
LRFD Bridge Design
TEH
3/5
287
Reinforced Concrete Substructures
PROBLEM 9.3
TEH
4/5
288
PROBLEM 9.3
LRFD Bridge Design
TEH
5/5
289
Reinforced Concrete Substructures
PROBLEM 9.4
TEH
1/5
290
PROBLEM 9.4
LRFD Bridge Design
TEH
2/5
291
Reinforced Concrete Substructures
PROBLEM 9.4
TEH
3/5
292
LRFD Bridge Design
PROBLEM 9.4
TEH
4/5
PROBLEM 9.4
TEH
5/5
293
Reinforced Concrete Substructures
PROBLEM 9.5
TEH
1/4
294
PROBLEM 9.5
LRFD Bridge Design
TEH
2/4
295
Reinforced Concrete Substructures
PROBLEM 9.5
TEH
3/4
296
PROBLEM 9.5
LRFD Bridge Design
TEH
4/4
Response 2000 results from “CEE4380​-P09​-05​​.rsp”:
The strain in the concrete at the tension face is 0.00298. Response 2000 shows
that the strain in the reinforcement is εt = 0.00275.
297
Reinforced Concrete Substructures
PROBLEM 9.6
TEH
1/2
298
PROBLEM 9.6
LRFD Bridge Design
TEH
2/2
The strain at the extreme tension face is 0.00348. From the response 2000 data,
the strain at the steel closest to the tension face is εt = 0.00301
299
Reinforced Concrete Substructures
PROBLEM 9.7
TEH
1/1
300
PROBLEM 9.8
LRFD Bridge Design
TEH
1/2
301
Reinforced Concrete Substructures
PROBLEM 9.8
TEH
2/2
302
LRFD Bridge Design
9.14 EXERCISES
E9.1.
For the pile bent described in Problem 9.3, estimate the nominal flexural resistance using (a) the approximate method in AASHTO and (b)
a cross-section analysis program. Also determine the maximum strain
which could be used in a pushover analysis based on buckling of the
tube wall.
E9.2.
An 8 ft diameter drilled shaft is subjected to the Strength limit state loading
given below. The shaft has forty #14 longitudinal bars and #6 hoops spaced
at 12 inches on center. Material properties are: f y = 60 ksi, f’c = 4 ksi. The
distance from the outer surface of the shaft to the center of the longitudinal
bars is 4.25 inches. Determine whether the shaft satisfies Strength limit
state requirements for the loading condition given.
• Pu = 4,675 kips, axial compression
• Mu = 22,540 ft-kips, moment
• Vu = 644 kips, shear
E9.3.
A 14 inch square prestressed concrete pile embedded 12 inches into a pile
cap constructed from 3,000 psi concrete receives a Strength limit state load
of 220 kips. Determine the minimum total thickness of the pile cap required
based on two-way shear requirements. The pile in question is at the corner
of the pile cap. The distance from the edge of the cap to the center of the
pile is 1 ft 6 inches.
E9.4.
A spread footing supports a column with Strength limit state axial load,
Pu = 2,300 kips, at the base of the column. The footing is 12 ft × 12 ft in
plan and is 5 feet thick. Moments at the Strength limit state acting in unison with the given axial load are Mux = 7,475 ft-kips and Muz = 6,400 ft-kips.
Determine the maximum soil pressure exerted by the footing (a) assuming a
uniform pressure distribution and (b) assuming a linearly varying pressure
distribution. Check eccentricity limits as well.
E9.5.
Determine the minimum and maximum pile axial loads. The factored loads
at the base of the column are:
• Pu = 800 kips
• Mux = 2,000 ft-kips
• Muz = 1,800 ft-kips
The 11 ft × 11 ft × 5.5 ft pile cap is shown in Figure E9.5. There is no pile
underneath the circular column in the center of the pile cap.
Reinforced Concrete Substructures
FIGURE E9.5
303
Exercise E9.5.
E9.6.
A rectangular pier column (f’c = 4 ksi, f y = 60 ksi) is subjected to the following loads at the Strength limit state:
• Pu = 1,877 kips
• Mux = 3,250 ft-kips
• Muz = 5,126 ft-kips
The cross section is shown in Figure E9.6. Use the biaxial flexure approximate solution to determine if the column satisfies the Strength limit state
resistance requirements.
304
FIGURE E9.6
LRFD Bridge Design
Exercise E9.6.
10
Seismic Design of Bridges
This chapter is intended to provide a discussion of basic seismic design principles,
both force-based and displacement-based conventional design.
Seismic isolation bearings were discussed in Chapter 8, Section 8.3, and seismic
displacement capacity calculations were discussed in Chapter 9, Section 9.7. Those
discussions will not be repeated here, but the reader will find occasional reference
to those sections informative while going through the material in the current chapter. Additionally, Chapter 11 introduces additional material on seismic isolation of
bridges.
Seismic design of bridges may be accomplished using either (a) force-based
design in accordance with the LRFD BDS (AASHTO, 2020), or (b) displacementbased design in accordance with the LRFD GS (AASHTO, 2011).
Displacement-based design is generally accepted as the more appropriate engineering approach, given that seismic design involves loading beyond the elastic
range.
Section 2.8 of this text presented a discussion on the development of seismic
loads in terms of a design response spectrum and the assignment of a Seismic Zone
(LRFD BDS) or a Seismic Design Category (LRFD GS). Each design method will
now be discussed.
This chapter deals with conventional design, based on either (a) plastic hinge
development in substructure columns or (b) ductile cross-frames with essentially
elastic substructures. For discussion of a third seismic design strategy, seismic isolation, refer to Chapter 11 of this text.
While typical design procedures incorporate linear response spectrum analysis
techniques, design by nonlinear response history analysis is also permitted. Response
history analysis requires the development of not only a design response spectrum,
but also a suite of appropriately selected and modified ground motion record pairs.
Hence, this Chapter also includes a detailed discussion of ground motion selection
and modification for inelastic response history analysis.
Design ground motion in the current AASHTO LRFD BDS is based on geometric mean, uniform hazard spectra. The likelihood is high that the near future will
find changes in design earthquake ground motion for bridges. For example, ASCE
7-16 for buildings now adopts risk-targeted (rather than uniform hazard), maximum
direction (RotD100, rather than geometric mean) ground motion as the design basis.
Additionally, design response spectra have typically been generated using three control points (AS, SDS, and SD1) to define the entire design response spectrum. These
data are available from the USGS. But the future may find bridge engineers using
a 22-point (rather than a three-point) design response spectrum, with site condition
effects explicitly incorporated into the mapped values of spectral acceleration. Such a
DOI: 10.1201/9781003265467-10
305
306
LRFD Bridge Design
tool is currently available, in beta version, at the USGS (https://earthquake​.usgs​.gov​/
ws​/designmaps​/nehrp​-2020​.html).
It is critical that bridge engineers maintain design procedures aligning with the
most up-to-date seismic loading definitions.
10.1 FORCE-BASED SEISMIC DESIGN BY THE LRFD BDS
In force-based seismic design by the LRFD BDS, a force-reduction factor is first
established. This requires that the owner specifies an importance category for each
particular bridge project. These categories include (a) critical, (b) essential, and (c)
other. These categories are likely to change in the near future, but, as of May 2021,
are as specified herein. For design by response spectrum analysis, force-reduction
(R) factors are summarized in Table 10.1. For design by inelastic response history
analysis, R = 1.0. With inelastic response history analysis, inelasticity is directly and
explicitly modeled.
For the connection of ductile columns or pile bent piles to the cap and foundation,
it is permitted to use capacity protection procedures in which overstrength capacity
of ductile elements is determined to establish the maximum load which can physically be transmitted to the connection or connected element. The capacity protection
method is preferred and may be used in lieu of the R-factors for connections.
Once determined by an acceptable analysis technique, elastic seismic forces are
reduced by the factor, R, and combined for orthogonal, horizontal effects as follows
to account for bi-directional ground motions.
• 100 percent of the design force in direction 1 combined with 30 percent of
the design force in direction 2
• 100 percent of the design force in direction 2 combined with 30 percent of
the design force in direction 1
TABLE 10.1
Force-Reduction (R) Factor for Force-based Seismic Design
Substructure
Critical
Essential
Other
Wall-type piers
1.5
1.5
2.0
Concrete pile bents, vertical piles only
Concrete pile bents, batter piles
Single-column bents
Steel and CFST pile bents, vertical piles only
Steel and CFST pile bents, batter piles
Multi-column bents
Connection, superstructure to abutment
Within-span expansion joints
Connection, columns, or pile bents to cap
1.5
1.5
1.5
1.5
1.5
1.5
0.8
0.8
1.0
2.0
1.5
2.0
3.5
2.0
3.5
0.8
0.8
1.0
3.0
2.0
3.0
5.0
3.0
5.0
0.8
0.8
1.0
Connection, columns, or piles to foundation
1.0
1.0
1.0
307
Seismic Design of Bridges
When non-ductile, capacity-protected elements are designed for the expected, overstrength capacity of the ductile elements, such combination is unnecessary. The
overstrength plastic shear must simply be resisted in any direction.
For bridges in Seismic Zone 1 with AS less than 0.05, the design force in any
direction is to be taken no less than 15 percent of the total vertical reaction due to
combined (a) permanent loads and (b) live load assumed to act concurrently with the
earthquake. For other bridges in Seismic Zone 1, the design force in any direction is
to be taken as not less than 25 percent of the total vertical reaction.
For Seismic Zone 2, capacity-protected, non-ductile elements are designed for a
force-reduction factor, R, equal to one-half of the value from Table 10.1, but no less
than 1.0. Once again, an alternative is to design for expected, overstrength capacity
of the ductile elements.
For Seismic Zones 3 and 4, hinging, ductile elements (usually the columns in a
multi-post bent, or the piles in a pile bent) are first designed using the reduced seismic forces. Capacity-protected, non-ductile elements are then designed for either (a)
unreduced elastic seismic forces (R = 1.0) or (b) the inelastic hinging forces. Inelastic
hinging forces are to be determined using resistance factors, ϕ = 1.3 for concrete or
ϕ = 1.25 for structural steel, to account for overstrength. This is an estimate of the
plastic shear for a bent or pier. The plastic shear is the largest load which may be
transmitted from a ductile element to the connected elements. Capacity design for
plastic hinging forces is to be greatly preferred over design based on unreduced seismic forces, in the author’s opinion.
Section 5.11 of the LRFD BDS contains specific seismic requirements for concrete elements. For details not covered here, the reader is referred to that section.
Requirements outlined here are generally those applicable to Seismic Zones 3 and
4, as those requirements are more fully developed and appropriate for the seismic
design of bridges, in the author’s opinion.
For circular columns and piles, Equation 10.1 provides the required volumetric
spiral or seismic hoop reinforcement for plastic hinge regions. The core diameter, dc,
is measured to the outside diameter of the spiral or hoop. The pitch, s, is measured
vertically to the center of the spiral or hoops. Asp is the cross-sectional area of the
spiral or hoop bar. The specified minimum yield strength, f yh, is that for the spiral
or hoop bar, but not necessarily the same as that for longitudinal bars. The specified
concrete strength, f’c, is used for the calculations, not the expected concrete strength,
f’ce. Equation 10.2 from Section 5.6.4.6 of the LRFD BDS for non-seismic confinement requirements, should always be checked as well. The core area, Ac, is based on
the diameter to the outside of the spiral or hoop as well.
Equations 10.3 and 10.4 provide the required confinement reinforcement, Ash,
within a spacing equal to s, for rectangular piles and columns. The dimension, hc, is
measured to the outside of the transverse bars in the direction of loading. The spacing, s, is to exceed neither (a) 4 inches, nor (b) one-quarter of the least cross-sectional
dimension.
rs =
4 Asp
f¢
³ 0.12 c
dc s
f yh
(10.1)
308
LRFD Bridge Design
rs =
4 Asp
æA
ö f¢
³ 0.45 ç g - 1 ÷ c
dc s
è Ac
ø f yh
Ash ³ 0.12shc
fc¢
f yh
(10.2)
(10.3)
æA
ö f¢
Ash ³ 0.30shc ç g - 1 ÷ c
è Ac
ø f yh
(10.4)
For either round or rectangular columns and piles, the yield strength of the transverse reinforcement, f yh, is the minimum specified value for the reinforcement used,
but is not to exceed 75 ksi.
10.2 DISPLACEMENT-BASED SEISMIC DESIGN BY THE LRFD GS
Displacement-based seismic design involves (a) identification of ductile elements
to be designed and detailed for inelastic behavior, (b) ensurance that the displacement demand on each ductile element is less than its displacement capacity, and (c)
capacity-protection of elements intended to remain essentially elastic during strong
ground shaking.
Displacement ductility is defined as the maximum displacement experienced by
an element divided by the yield displacement of the element. For pile bent design
by the displacement-based provisions of the LRFD GS, the displacement ductility
demand, μD, is not to exceed 4. For single-column bents, μD is limited to 5, and for
multi-columns bents with above-ground hinging, μD is limited to 6.
Displacement capacity has been discussed in Section 9.7 of this text. Displacement
demand may be determined by (a) equivalent static analysis with displacement
amplification for short-period structures, (b) multi-mode elastic response spectrum
analysis with displacement amplification for short-period structures, or (b) inelastic response history analysis. Orthogonal component combination for methods (a)
and (b) are based on the 100-30-30 rule, while orthogonal component interaction is
inherently accounted for in inelastic response history analysis, when three-dimensional modeling is adopted. Refer to the LRFD GS, Section 4.2, for guidance on
permissible analysis methods for various bridge complexity levels.
For equivalent static and elastic response spectrum methods, short-period displacement amplification is required for natural modes having periods less than or equal to
1.25TS. See Section 2.8 on earthquake loading for the definition of TS. The required
amplification, Rd, of displacement demand is given by Equation 10.5. Design response
spectra are typically based on damping equal to 5 percent of critical. In rare cases
for which larger damping is appropriate, Equation 10.6 is provided in the LRFD GS
for modification of the design response spectrum across all periods by the factor, R D.
æ
1
Rd = ç 1 è mD
ö æ 1.25TS
֍
øè T
ö 1
÷+ m
ø
D
(10.5)
309
Seismic Design of Bridges
æ 0.05 ö
RD = ç
÷
è x ø
0.40
(10.6)
For Equation 10.5, μD is equal to 2 for Seismic Design Category B, 3 for Seismic
Design Category C, and determined by analysis for Seismic Design Category D.
For Seismic Design Category D, μD may conservatively be taken to be equal to 6 in
Equation 10.5.
Second-order PΔ effects may be ignored in structural analysis for earthquake
loading whenever Equation 10.7 is satisfied for concrete substructures, or Equation
10.8 for steel substructures. The unfactored dead load is Pdl. Mp is the idealized
plastic moment determined using expected, rather than specified minimum, material properties. Mn is the nominal moment capacity based on nominal properties.
Presumably, Δr is the first-order displacement between points of contraflexure and
plastic hinging (approximated by one-half of the total displacement for a rigid frame
with hinges at top and bottom of columns).
Pdl D r £ 0.25M p
(10.7)
Pdl D r £ 0.25M n
(10.8)
Equation 10.9 for analytical plastic hinge length, LP, is appropriate for reinforced
concrete columns framing into oversized shafts, footings, cased shafts, and bent
caps. In the equation, L is the distance from the point of maximum moment to the
point of contraflexure, not necessarily equal to the clear column height. The diameter, dbl, is that for the longitudinal bars in the column. For reinforced and prestressed
concrete piling, and for cast-in-drilled-hole shafts, Equation 10.10a provides the analytical plastic hinge length. H’ is the distance from ground surface to the point of
above-ground contraflexure. D* is the diameter for a circular member, or otherwise
the cross-sectional dimension in the direction of loading. For concrete-filled steel
tube pipe piles of diameter D, Equation 10.10b is specified in the LRFD GS.
LP = 0.08L + 0.15 f ye dbl ³ 0.30 f ye dbl
(10.9)
LP = 0.1H ¢ + D* £ 1.5D*
(10.10a)
LP = 0.1H ¢ + 1.25D £ 2.0 D
(10.10b)
The plastic hinge region for reinforced concrete columns and piles, within which
enhanced lateral confinement reinforcement is to be provided, is taken as the
greater of:
• 1.5 times the dimension in the direction of loading
• the region where moment demand exceeds 75 percent of the plastic moment
• the analytical plastic hinge length, LP
310
LRFD Bridge Design
Minimum support lengths at expansion joints are determined from Equation 10.11 for
Seismic Design Category A. L is the length, in feet, of the bridge to the adjacent expansion joint or the end of the bridge. N is the minimum required support length, in inches.
S is the skew of the deck to substructure alignment, in degrees. H is the average height,
in feet, of columns supporting the bridge deck at piers from abutment to abutment.
(
N = ( 8 + 0.02 L + 0.08 H ) 1 + 0.000125S 2
)
(10.11)
For Seismic Design Categories B and C, the length given by Equation 10.11 is to be
increased by a factor of 1.5.
For Seismic Design Category D, the required support length is a function of the
displacement demand from the structural analysis (Δeq), and is given by Equation
10.12. For single-span bridges in Seismic Design Category D, the support length is
taken to be not less than 1.5 times the value from Equation 10.11. The skew effect
multiplier in Equation 10.12 is different from that in Equation 10.11. It is not clear
if this is intentional in the LRFD GS. Nonetheless, the equations presented here are
taken from the LRFD GS.
(
)
N = ( 4 + 1.65D eq ) 1 + 0.00025S 2 ³ 24
(10.12)
For structural analysis, abutment longitudinal stiffness (Keff ) and strength (Pp) may
be ignored, with intermediate piers designed to resist all seismic effects, or may be
included (when permitted by the owner) as given in Equations 10.13 and 10.14.
Pp = H wWw p p
(10.13)
Pp
Fw H w
(10.14)
K eff 1 =
•
•
•
•
•
•
Keff = abutment stiffness, kips/ft
Pp = passive capacity, kips
pp = presumptive passive pressure, ksf
Hw = backwall height, ft
Ww = backwall width, ft
Fw = factor ranging from 0.01 (dense sand) to 0.05 (compacted clay)
Unless special attention, beyond that normally provided for backfill, is introduced
into the plans and specifications, 70 percent of the presumptive passive pressure, pp,
should be used in the analysis. Refer to the LRFD GS, Section 5.2.3.1, for additional
information. Presumptive pressure, if used in lieu of detailed analysis procedures,
requires that the backfill be compacted to a dry density greater than 95 percent of the
maximum, and is specified in the LRFD GS as follows:
• for cohesionless, non-plastic backfill, pp may be taken equal to 2Hw/3 ksf
per foot of wall length.
Seismic Design of Bridges
311
• provided that the estimated undrained shear strength is greater than 4 ksf,
pp for cohesive backfill may be taken to be equal to 5 ksf.
Equation 10.14 provides an initial stiffness estimate, which may be used in a
response spectrum or response history analysis in cases for which abutment stiffness
and strength is to be used to resist seismic forces. From the initial structural analysis,
the longitudinal forces at the abutments will be available. Should this force exceed
Pp, the initial spring stiffness should be softened and the analysis re-run. The iterative procedure should be performed until the assumed stiffness is consistent with the
computed stiffness.
10.3 CAPACITY DESIGN PRINCIPLES
Displacement-based and force-based design both require attention to capacity-design
principles. Ductile elements are detailed to ensure inelastic deformation capacities in
excess of the estimated required deformation during strong ground shaking. Other
elements are designed for a higher force level, thus ensuring their essentially elastic
behavior during the event. For bridge substructures, when column plastic hinging
is the design strategy, overstrength plastic shear calculations are made and all other
elements are designed for the expected, overstrength plastic shear. Some of these
capacity design provisions from the AASHTO LRFD GS are summarized here.
Readers are referred to the LRFD GS for a complete treatment of these principles
and requirements.
Plastic hinging forces are to be computed using expected (not minimum) material strengths with an additional over-strength factor. The over-strength factor, λmo, is
determined as follows:
• λmo = 1.4, reinforced concrete hinging columns with A 615 reinforcing
• λmo = 1.2, reinforced concrete with A 706 reinforcing
• λmo = 1.2, structural steel hinging columns
Expected material strengths are f’ce = 1.3f’c for concrete and 68 ksi for the yield stress,
f ye, of steel reinforcing bars, whether A615 Grade 60 or A706 Grade 60. Expected
tensile strength, fue, is 95 ksi for both bar specifications. For hinging columns in
Seismic Design Category D, A706 reinforcement is required in lieu of A615. Strainhardening effects are to be incorporated into the reinforcing steel material model.
Strain limits in reinforcement are summarized below, with recommended ultimate tensile strains on the order of two-thirds to three-fourths of actual minimum
values for added safety. The onset of strain hardening is defined by εsh, and the ultimate tensile strain by εRsu.
For the onset of strain hardening:
• εsh = 0.0150, A 615 and A 706 bars, #3–#8
• εsh = 0.0125, A 615 and A 706 bars, #9
• εsh = 0.0115, A 615 and A 706 bars, #10–#11
312
LRFD Bridge Design
• εsh = 0.0075, A 615 and A 706 bars, #14
• εsh = 0.0050, A 615 and A 706 bars, #18
For bar fracture:
•
•
•
•
εRsu = 0.090, A 706 bars, #4–#10
εRsu = 0.060, A 615 bars, #4–#10
εRsu = 0.060, A 706 bars, #11–#18
εRsu = 0.040, A 615 bars, #11–#18
Limiting compressive strains in the concrete core are typically determined using the
Mander model for confined concrete (see Section 9.7 of this text) and depend on the
amount of transverse reinforcement provided in the form of hoops, spirals, or rectangular ties. Core compressive strains at in-ground plastic hinges, when permitted,
should be limited to 0.02.
Determination of the plastic shear for a multi-column bent or pier is an iterative procedure. Problem 10.2 provides a detailed example, including confinement
calculations, displacement demand calculations, displacement capacity calculations,
and plastic shear calculations. Section analysis software is a requirement for such
analyses.
Capacity design provisions for footings may be found in the AASHTO LRFD GS,
Sections 6.4.5, 6.4.6, and 6.4.7. For a footing of length L in the direction of loading,
column dimension Dc in the direction of loading, and depth Hf, Equation 10.15 must
be satisfied in order to consider the footing or pile cap rigid. Otherwise, non-standard
analyses are required to determine pressure distribution for a footing, or pile loads
for a pile cap.
Equation 10.16 is the overturning analysis requirement for a spread footing. Mpo
is the overstrength plastic moment of the column and Vpo is the overstrength plastic
shear. The axial force associated with the overstrength plastic hinging forces is Pu and
the resistance factor, ϕ, for overturning, is 1.0. The nominal bearing capacity of the
supporting material is qn. The footing width, perpendicular to the direction of loading, is B. The footing width effective in resisting flexure and shear, beff, is given by
Equation 10.18. The column dimension perpendicular to the direction of loading is Bc.
( L - Dc ) £ 2.5
2H f
æ L-a ö
M po + Vpo H f £ f Pu ç
÷
è 2 ø
a=
Pu
qn B
beff = Bc + 2 H f £ B
(10.15)
(10.16)
(10.17)
(10.18)
313
Seismic Design of Bridges
For additional footing and pile cap requirements not covered here, refer to the LRFD
GS, Sections 6.3 and 6.4.
Capacity design provisions for non-integral bent caps extensive and are found in
Sections 8.12 and 8.12 of the LRFD GS. The reader is referred to these provisions for
detailed design requirements.
10.4 GROUND MOTION SELECTION AND MODIFICATION
FOR RESPONSE HISTORY ANALYSIS
For structural design by response history analysis, it becomes necessary to select a
suite of ground motion records. Typical suite sizes range from as few as three record
pairs to as many as 11 or more. In fact, given the emphasis on performance-based
design likely to occur in the near future, as many as 30 or 40 record pairs (or more)
could be a necessary minimum in research when estimates of response variability to
ground shaking are needed.
Ground motion selection requires careful attention to several factors. An excellent reference for the process is found in NIST-GCR-11-917-15 (NEHRP Consultants
Joint Venture, 2011). Engineers and researchers working on ground motion selection
would be well served in consulting this freely available digital document.
Some of the parameters involved in selecting candidate ground motions for structural analysis include, in descending order of importance (in the author’s opinion), at
least for non-subduction earthquakes:
1.
2.
3.
4.
5.
match to spectral shape
magnitude
recording station site characterization
distance
fault type
Given that match to spectral shape is a critical factor in whether or not a particular ground motion should be considered for a given site, some means of measuring
match to spectral shape is necessary. Two proposed measures of match to spectral
shape are mean-square-error (MSE) and DRMS. MSE and DRMS are given in Equations
10.19 (Pacific Earthquake Engineering Research Center, 2010) and 10.20. Notice
that MSE has a scale factor, f. Equation 10.21 provides an expression for computing
the scale factor, f, which minimizes MSE over a specified range of periods. This is
the scale factor used in amplitude scaling of ground motion records discussed in the
section on ground motion modification. So MSE can be computed pre-scaling by
inserting f = 1 in the equation for MSE, and post-scaling by inserting the computed
scale factor for f in the equation for MSE. The post-scaled MSE is the appropriate
value for assessing candidate records. The weights, w(Ti), at each period are typically
taken to be equal to 1 for all periods in the range of interest.
{
}
å w ( Ti ) × ln éë SATARGET ( Ti ) ùû - ln éë f × SAGM ( Ti ) ùû
MSE =
å w ( Ti )
2
(10.19)
314
LRFD Bridge Design
DRMS =
1
N
N
å
i =1
æ ( SAGM )Ti ( SATAR )Ti
ç
ç PGAGM
PGATAR
è
é
SATARGET ( Ti ) ù
å ê w ( Ti ) × ln
ú
SAGM ( Ti ) úû
ê
ln f = ë
å w ( Ti )
ö
÷
÷
ø
2
(10.20)
(10.21)
A knowledge of characteristic magnitude and distance combinations may be obtained
for a given site by disaggregating the seismic hazard, available at the USGS online
Unified Hazard Tool (https://earthquake​.usgs​.gov​/ hazards​/interactive/).
Once a set of candidate records (perhaps 100 or more) has been established, it
is necessary to have a means of (a) selecting the most appropriate records and (b)
modifying the records. While many candidate records from a single event may be
included, it is customary (and sometimes required) that no more than three or four
records from any single event be included in the final suite.
There are at least three methods for modifying ground motion records for use in
structural analysis:
1. amplitude scaling
2. spectral matching in the time domain
3. spectral matching in the frequency domain
Amplitude scaling is typically the preferred method for ground motion modification.
The accelerations at each time step in the record are all multiplied by an appropriate
factor, such as that computed from Equation 10.17. The time scale is not adjusted in
any fashion. Frequency content and pulse-type character of the ground motion are
retained.
The PEER Ground Motion Database (Pacific Earthquake Engineering Research
Center, 2014) includes an excellent online tool for ground motion selection with subsequent amplitude scaling. Software for ground motion scaling includes SigmaSpectra
(Albert Kottke), which is freely available, and SeismoSelect, for which free educational licenses are available.
SigmaSpectra (https://github​.com​/arkottke​/sigmaspectra) offers the advantage of
permitting the user to specify a target log-based standard deviation in addition to a
target pseudo-spectral acceleration (PSA) spectrum. ASCE 7-16 (Section 21.2.1.2)
suggests a log-based standard deviation equal to 0.60 across all periods of interest.
This presumably preserves record-to-record variability with a small subset of all
available ground motions, and could be used to design to achieve responses greater
than the median (by some number, say 1, of standard deviations). Ground motion
models may also be used to establish target, log-based variability.
.seismosoft​
.com) provides another alternative for ground
SeismoSelect (www​
motion scaling. Included in the software are various options for target basis (RotD100,
Seismic Design of Bridges
315
GeoMean, etc.) as well as for ground motion database sources (PEER, ESMD, etc.).
SeismoSelect also has features which enable the user to generate code-based target
spectra for many different specifications.
For spectral matching in the time domain, SeismoMatch (SeismoSoft, 2020b)
adds wavelets to an accelerogram to produce a new accelerogram to which the PSA
response spectrum matches, as closely as possible, the target spectrum. Modification
is performed directly on the accelerogram in the time domain.
SeismoArtif (SeismoSoft, 2020a) has multiple capabilities, among which is spectral matching in the frequency domain. The Fourier spectrum for a record is first
computed and compared to a Fourier spectrum generated from the target PSA spectrum. Adjustment to the ground motion Fourier spectrum is made to more closely
match that of the target Fourier spectrum. The modified Fourier spectrum is then
converted back to a new accelerogram.
Care must be taken in spectral matching, whether time-domain-based or frequency-domain-based, to ensure that realistic (if somewhat subjective) ground
motions are produced in the matching process. Any pulse-type character for nearfault effects required by a governing code or specification may be significantly
altered when spectral matching is employed, but is preserved when amplitude scaling is used.
In addition to modification of real ground motion records, artificial and synthetic
records may, at times, be useful. Real records, modified by amplitude scaling, are the
preferred choice, in the author’s opinion. Nonetheless, a brief discussion of artificial
and synthetic record options is appropriate.
Physics-based models exist for generating synthetic ground motions. SeismoArtif
(SeismoSoft, 2020a) has this capability, in addition to those previously mentioned.
Near-fault and far-field options are available. Site conditions and tectonic environment (intraplate vs. plate boundary) are specified by the user. Both synthetic accelerogram generation/adjustment and artificial accelerogram generation/adjustment
are available. In the context adopted here, synthetic accelerograms are those based
on physics-based models. Artificial accelerograms are those generated as a random
signal and bounded with a specified enveloping function.
A simple model for synthetic record generation is available through the State
University of New York (SUNY) program RSCTH. Required input is minimal,
including moment magnitude, epicentral distance, target spectrum parameters, and
tectonic regime (Eastern US or Western US). The program is DOS based.
For the target response spectrum, it is necessary to assign a period range of interest. Design specifications vary in defining this period range. While appropriate for
buildings, ASCE 7-16 may be the preferable standard available in defining an appropriate period range of interest. ASCE 7-16 defines the period range of interest as
follows.
• The period range of interest should have an upper bound greater than or
equal to twice the larger of the two fundamental periods in the two orthogonal, horizontal directions. In the opinion of the author, this should be
316
LRFD Bridge Design
extended to 2.5 times the larger period, given that the effective period of an
elastic-perfectly-plastic structure having a displacement ductility demand
of 6 (the largest value permitted for conventional design in the AASHTO
LRFD GS) is equal to the elastic period multiplied by the square root of 6
(i.e., 2.45).
• The period range of interest should have a lower bound equal to the smaller
of (a) the period required to achieve at least 90% mass participation in each
horizontal, orthogonal direction and (b) 20% of the smallest fundamental
period in the two horizontal, orthogonal directions.
For ground motion selection requirements, AASHTO is relatively silent. However,
the FHWA retrofit manual for bridges (Buckle, et al., 2006) does contain a few
requirements. These are summarized below.
• Either three ground motions or seven ground motions may be used in the
analysis. If three ground motions are used, the maximum response of the
three should be used for design. If seven are used, the average response may
be taken as the design value.
• The suite mean spectrum should not fall below the target by more than 15%
over the period range of interest, and the average ratio of suite mean to target over the entire period range of interest should be at least 1.0.
• Amplitude scaled or spectrally matched records are permissible.
• For near-fault sites, ground motions are to be rotated to fault-normal (FN)
and fault-parallel (FP) orientations for application in structural analysis.
Although not strictly required by current AASHTO provisions, it seems wise to adopt
ground motion suites with no fewer than eleven record pairs for bridge design, given
that the ASCE provisions for suite size are more developed than those in AASHTO.
Figure 10.1 depicts an example of an appropriately generated ground motion suite.
The plot clearly illustrates that more stringent criteria than those found in the FHWA
retrofit manual or in AASHTO are generally satisfied by the suite of record pairs.
Such criteria may include each of the following, depending on the design specification adopted.
• For amplitude scaling, record pairs should be scaled such that the suite
mean PSA spectrum generally matches or exceeds the target spectrum over
the period range of interest. This may be quantitatively evaluated by averaging the suite-mean PSA to target PSA ratio over the period range of interest.
• For amplitude scaling, record pairs should be scaled such that the suite
mean PSA spectrum does not fall below 90% of the target spectrum at any
period within the range of interest.
• For amplitude scaling, record pairs should be scaled such that the suite
mean PSA spectrum does not exceed 130% of the target spectrum at any
period within the range of interest.
Seismic Design of Bridges
FIGURE 10.1
317
Example ground motion suite spectra.
• The target, log-based standard deviation on PSA should equal 0.60 over
the period range of interest. Ground motion models may also be useful in
establishing target, log-based standard deviation.
In Figure 10.1, the period range of interest is [0.0–3.0] seconds. The minimum
PSASuite/PSATarget is 0.935 (greater than 0.9 required). The average PSASuite/PSATarget
is 1.000 (greater than or equal to 1.0, and preferable very near 1.0). The maximum
PSASuite/PSATarget is 1.028 (less than 1.3 as required). Over the period range of interest, the average σlnPSA is 0.599 (very near the target 0.60).
10.5 SUBSTITUTE-STRUCTURE METHOD (SSM) ANALYSIS
The substitute-structure method (SSM) for seismic analysis is often a valuable tool
in estimating seismic displacement demands. Although the final design for critical
structures is likely to require nonlinear response history analysis for final design
details, SSM analysis has been shown to provide reliable seismic displacement
estimates (Huff and Pezeshk, 2016). The method, originally proposed by Gulkan
and Sozen (1974), is the basis for the preliminary design of isolation devices in the
AASHTO GS ISO (AASHTO, 2014).
SSM analysis is a response spectrum-based procedure and requires no ground
motion selection. Only the design response spectrum is required for loading definition. To complete the analysis, secant stiffness (KEFF ) at maximum displacement
is used to compute an effective period (TEFF, Equation 10.22). Figure 10.2 depicts
the essential parameters in an assumed bi-linear force displacement relationship.
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LRFD Bridge Design
FIGURE 10.2 Bi-linear force-displacement in the SSM analysis.
Models for (a) added damping and (b) response modification due to added damping
are required as well. Displacement ductility, μD, is defined by Equation 10.23.
Various damping (ξEFF ) models include those proposed by Priestley et al. (2007).
Equation 10.24 has been proposed as valid for reinforced concrete bridge substructures under inelastic displacement demands. Equation 10.25 is the theoretical elastic plus hysteretic damping for any bilinear oscillator. For steel framed buildings,
Equation 10.26 was proposed by Priestley et al. (2007).
Models for response modification (Rξ) as a result of added damping include
Equation 10.27 for non-pulse type ground motion and Equation 10.28 for pulse-type
ground motion, both from Euro Code 8. The equivalent of Equation 10.29 is found
in the AASHTO GS ISO (AASHTO, 2014) for seismic isolation devices. A model
which depends on both effective damping and significant duration (D5-95) has been
developed as well (Stafford et al., 2008), given by Equation 10.30.
The structure is treated as a single-degree-of-freedom (SDOF) oscillator in the
SSM. From the design response spectrum, PSATEFF is the pseudo-spectral acceleration at the effective period of the SDOF. The displacement demand is given by
Equation 10.31.
Careful observation of the equations will make it clear that the displacement
demand, DMax, depends on the effective damping, ξEFF, which depends on DMax.
Hence, the iterative nature of the SSM becomes evident. Hence, depending on the
type of problem, it may be necessary to assume an initial value for either μD, DMax,
or ξEFF, performing subsequent iterations until the calculated results agree with the
assumption.
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Seismic Design of Bridges
TEFF = 2p
W
mD
= Ti
gK EFF
1 + am D - a
mD =
DMax
Dy
x EFF = 0.05 + 0.444
x EFF = x EL +
(10.23)
mD - 1
pm D
(10.24)
2 ( m D - 1) (1 - a )
pm D (1 + am D - a )
x EFF = 0.05 + 0.577
(10.26)
0.50
æ
ö
0.07
Rx = ç
÷
è 0.02 + x EFF ø
æ 0.05 ö
Rx = ç
÷
è x EFF ø
(10.25)
mD - 1
pm D
æ
ö
0.10
Rx = ç
÷
è 0.05 + x EFF ø
Rx = 1 -
(10.22)
(10.27)
0.25
(10.28)
0.30
³ 0.588
-0.631 + 0.421 ln (x EFF ) - 0.015 ln (x EFF )
ìï - é ln ( D5- 95 ) - 2.0047 ù üï
û
1 + exp í ë
ý
0
930
.
îï
þï
2
(10.29)
2
(10.30)
æT ö
DMax = ( PSATEFF × g ) ç EFF ÷ × Rx
(10.31)
è 2p ø
In approximate analyses using SSM techniques, it may at times be desirable to estimate effects due to plan torsional irregularities. The method outlined in ASCE 7-16,
Section 17.5.3.3, can be useful for such estimates. Equation 10.32 provides an estimate for displacement amplification due to plan torsional offset of the center of mass,
often assumed to be 5% or more of the plan dimension. Eccentricity, e, is defined
to be the distance between the centers of mass and rigidity. The plan dimensions
are b and d. The distances from the center of mass to each isolator unit along two
320
LRFD Bridge Design
perpendicular axes are x and y. The distance from the center or rigidity to an isolator
in question is z.
é
z
12e ù
D¢Max = DMax ê1 + 2 × 2
³ 1.15DMax
P
b
+ d 2 úû
T
ë
PT =
(
2
2
1 å x +y
r
N
r=
)
b2 + d 2
12
(10.32)
(10.33)
(10.34)
10.6 SHEAR RESISTANCE AT THE EXTREME EVENT LIMIT STATE
For reinforced concrete columns subject to large seismic shear, conventional resistance models may prove unreliable. Multiple methods exist for the evaluation of
shear resistance at the Extreme Event limit state for columns which form plastic
hinges during strong ground shaking.
The AASHTO LRFD GS adopts Equation 10.35 for design shear resistance within
the plastic hinge region and recommends a resistance factor ϕ = 0.90 rather than the
value of 1.0 used for most Extreme Event limit states. The same equation may be
used for ductile columns outside the plastic hinge region by setting α’ equal to 3. The
concrete shear stress, vc, is calculated from Equation 10.37 if the factored axial load,
Pu, is compressive. Otherwise, vc = 0. Equations 10.38, 10.39, and 10.40 are applicable for concrete resistance of round columns. Equations 10.41, 10.42, and 10.43 are
applicable for concrete resistance of rectangular columns. For the shear resistance
provided by reinforcement, Equation 10.44 is for round columns and Equation 10.45
is for rectangular columns.
fVn = 0.90 (Vc + Vs )
(10.35)
Vc = vc ( 0.80 Ag )
(10.36)
ìï 0.11 fc¢
æ
P ö
vc = 0.032a ¢ ç 1 + u ÷ fc¢ £ min í
è 2 Ag ø
îï0.047a ¢ fc¢
(10.37)
a¢ =
fs
+ 3.67 - m D
0.15
fs = r s f yh £ 0.35
rs =
4 Asp
sD¢
(10.38)
(10.39)
(10.40)
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Seismic Design of Bridges
a¢ =
fw
+ 3.67 - m D
0.15
fw = 2 r w f yh £ 0.35
(10.43)
p æ nAsp f yh D¢ ö
÷ £ 0.25 fc¢ ( 0.80 Ag )
2 çè
s
ø
(10.44)
Av f yh d
£ 0.25 fc¢ ( 0.80 Ag )
s
(10.45)
Vs =
•
•
•
•
•
•
•
•
•
•
•
•
•
(10.42)
Av
sb
rs =
Vs =
(10.41)
Ag = gross cross-sectional area, in2
Pu = factored compressive force, kips
Asp= area of spiral or hoop bar, in2
s = pitch of spiral, hoops, or ties, inches
D’ = core diameter measured to center of spiral or hoop, inches
Av = total area of shear reinforcement in the direction of loading, in2
b = width of rectangular member perpendicular to direction of loading,
inches
f yh = nominal yield stress of transverse reinforcement, ksi
f’c = nominal compressive strength of concrete, ksi
μD = local displacement ductility demand for the member in question
α’ = adjustment factor
n = number of individual interlocking cores with spiral or hoop bars
d = effective depth in the direction of loading, inches
The effective depth, d, for rectangular members is measured from the extreme compression face to the centroid of the tensile reinforcement.
For Seismic Design Category B, shear resistance within the plastic hinge zone
is to be determined using μD = 2. For Seismic Design Category C, shear resistance
within the plastic hinge zone is to be determined using μD = 3. For Seismic Design
Category D, shear resistance within the plastic hinge zone is to be determined using
μD as determined from analysis.
Minimum transverse reinforcement must be provided in accordance with the
following:
• for Seismic Design Category B, ρs ≥ 0.003 and ρw ≥ 0.002
• for Seismic Design Category C and D, ρs ≥ 0.005 and ρw ≥ 0.004
For members with interlocking cores and spiral or hoop reinforcement, the shear
reinforcement resistance is taken as the sum of the individual contributions from
each interlocking spiral or hoop.
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LRFD Bridge Design
For details which are not covered here, the reader is referred to the LRFD GS,
Section 8, and to the literature (Priestley et al., 2007) for additional shear models.
10.7 SOLVED PROBLEMS
Problem 10.1
The pile bent shown in Figure P10.1 is used as an intermediate pier in a
short-span continuous bridge. The CFST piles are 20-inch diameter with
5/8-inch wall thickness and are fabricated in accordance with ASTM A53
Grade B, Fy = 35 ksi. The concrete core has a 28-day compressive strength,
f’c = 3 ksi. For the condition shown, each of the five girder-bearing locations
has a reaction of 117 kips.
The cap is 4 ft by 4 ft in cross section. The lateral load shown represents
an incremental load to be applied at the superstructure center of gravity in
a pushover analysis.
Determine the plastic shear and the displacement capacity from pushover analysis. SeismoStruct will be used for the analysis.
Problem 10.2
A bridge pier is to be constructed at a seismic zone 4 (Seismic Design
Category D) site in the NMSZ. The geometry and data for the bridge piers
FIGURE P10.1
Problem P10.1.
Seismic Design of Bridges
323
FIGURE P10.2 Problem P10.2.
are given in Figure P10.2 and the ensuing discussion. The clear cover to
the transverse bars is 1.5 inches. Specified reinforcing yield strength is
f y = 60 ksi (A 706 Grade 60 bars) for both longitudinal and transverse reinforcement. Specified concrete strength is f’c = 3 ksi.
The bridge is a three-span structure consisting of 195-ft end spans
and a 230-ft central span, for a total bridge length of 620 feet. The
superstructure weight, including live load assumed to be present during
strong ground shaking, is 11.8 kips per foot. Seismic design data are as
follows:
PGA = 0.488 SS = 0.897 S1 = 0.227 Fa = 1.050 Fv = 3.32
•
•
•
•
Column diameter, D = 54 inches
Height to superstructure center of gravity, HCG = 47.5 feet
Clear column height, 2LC = 33.75 feet, spacing S = 16.875 feet
Initial column axial loads Pu = 1,290 kips (exterior columns), 1,067 kips
(center column)
• Displacement demand from response spectrum analysis = 4.64 inches
• Longitudinal bars are 30 #9, #6 hoops spaced at 6 inches
Determine the plastic shear, VP, and the displacement capacity from a pushover analysis, ΔCAP.
Problem 10.3
For the structure in Problem 10.2, select and scale a set of 14 ground motion
records for use in a nonlinear response history analysis of the structure.
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LRFD Bridge Design
The subsurface investigations indicate a Site Class D condition. The modal
magnitude, distance pair is M W, R = 7.75, 55 km. The fundamental periods
are 0.78 seconds in the longitudinal direction and 0.73 seconds in the transverse direction. Modal analysis results indicate that at least 90% of the total
mass is captured in each direction at a period of 0.55 seconds.
Problem 10.4
A wide, short-span bridge is modeled as a rigid block with the following
properties:
W = 2, 823 kips
K i = 178 kips/inch
The design seismic loading is identical to that for Problem 10.2. Use the
substitute structure method to determine a preliminary design yield force if
the displacement ductility is to be limited to no more than 6. Use an effective damping model appropriate for concrete substructures, and assume
pulse-type ground motion characteristics.
Problem 10.5
A 54-inch diameter column with 20 #10 longitudinal bars (A 706) and #6
hoops at 5 inches on center in the plastic hinging zone has a clear height of
37 feet and behaves as a rigid frame with fixity at both the top and bottom
of the column. Specified concrete strength for the column is 4 ksi and the
yield strength of the A 706 bars is 60 ksi.
• Determine the overstrength plastic shear of the column. The column
axial compression is Pu = 1,221 kips. Clear cover to the hoops is 2.5
inches.
• Determine the displacement ductility demand on the column if the displacement demand from a seismic analysis of the bridge is 12.6 inches
at the pier for which the column is located.
• Determine the displacement ductility capacity of the column in
question.
• Determine the adequacy of the column design in terms of seismic shear
demand.
• Establish the minimum square footing dimension which will satisfy
overturning requirements if the footing depth is 5.5 feet. Check the
square footing thus obtained for rigid footing criteria. The nominal
bearing resistance is qn = 20 ksf.
• Determine whether second-order effects need to be included in the
analysis which generate the displacement demand, given as 12.6 inches
for this problem. Assume that the dead load, Pdl, is 70% of the total
load, Pu.
Problem 10.6
A three-span bridge has a total length of 620 feet. One abutment is constructed on multiple rows of piles so as to be extremely stiff in the longitudinal
325
Seismic Design of Bridges
direction. This abutment is considered to be the fixed point on the superstructure. The other abutment incorporates an expansion joint to accommodate
seismic and thermal movements. The column height at both piers is 31 feet.
The seismic analysis indicates a displacement demand during strong ground
shaking of 17.7 inches in the longitudinal direction. Temperature loading
(TU) requirements at the expansion abutment necessitate the accommodation of 3.50 inches expansion and 3.50 inches of contraction.
Determine the required seat length at the expansion abutment if the
bridge is located in:
• Seismic Design Category A with no skew
• Seismic Design Category B with no skew
• Seismic Design Category C with no skew
• Seismic design Category D with no skew
• Seismic Design Category D with a 45-degree skew of substructures
PROBLEM 10.1
TEH
1/2
CFST24​.s​pf SeismoStruct Project File
• take f’ce = 1.3 × 3.0 = 3.9 ksi, LRFD GS 8.4.4
• take f ye = 1.5 × 35 = 52.5 ksi, expected yield for A53 from LRFD GS 7.3
• take λmo = 1.2, overstrength factor from LRFD GS 4.11.2
Apply 117 kip downward loads at each bearing location in the z-direction and a
1-kip incremental load at the center of gravity node in the y-direction and push
the pier to failure.
326
PROBLEM 10.1
LRFD Bridge Design
TEH
2/2
The plastic shear is found to be 788 kips and the ultimate displacement 8.63
inches. This could be misleading if the strain corresponding to this displacement
in the extreme piles is sufficient to cause buckling of the pile wall.
The shear per pile, with SeismoStruct overstrength included, is then 1.2 ×
(788/9) = 105 kips. In Chapter 9, Equation 9.94 gives a nominal shear resistance of
386 kips per pile. Hence, shear failure does not preclude flexural failure.
Finally, note that the maximum strain from the model, ε = 0.0013, is significantly less than that corresponding to tube buckling from Chapter 9, Equation
9.96 (0.00977).
The estimated displacement capacity of 8.63 is a realistic estimate since neither shear failure nor tube buckling occurs at this level of loading.
327
Seismic Design of Bridges
PROBLEM 10.2
TEH
1/4
328
PROBLEM 10.2
LRFD Bridge Design
TEH
2/4
329
Seismic Design of Bridges
PROBLEM 10.2
TEH
3/4
330
PROBLEM 10.2
LRFD Bridge Design
TEH
4/4
331
Seismic Design of Bridges
PROBLEM 10.3
TEH
1/2
332
PROBLEM 10.3
LRFD Bridge Design
TEH
2/2
333
Seismic Design of Bridges
PROBLEM 10.4
TEH
1/1
334
PROBLEM 10.5
LRFD Bridge Design
TEH
1/5
335
Seismic Design of Bridges
PROBLEM 10.5
TEH
2/5
336
PROBLEM 10.5
LRFD Bridge Design
TEH
3/5
337
Seismic Design of Bridges
PROBLEM 10.5
TEH
4/5
338
PROBLEM 10.5
LRFD Bridge Design
TEH
5/5
339
Seismic Design of Bridges
PROBLEM 10.6
TEH
1/1
340
LRFD Bridge Design
10.8 EXERCISES
E10.1.
For the prestressed girder alternative of the Project Bridge, estimate the
displacement capacity of the pier in the transverse direction (rigid frame
behavior) given the following data. Use hand calculations and base the
results on the initial column axial loads.
• column diameter = 42 inches
• clear column height = 29 feet
• distance from column base to superstructure center of gravity = 39 feet
• two-post pier, column spacing S = 18.5 feet
• f’c = 3 ksi
• f y = f yh = 60 ksi, A 706 bars
• longitudinal bars (per column), 14 #9
• transverse hoops (per column), #5 at 4 inches on center
• clear cover to hoops = 2 inches
• initial column loads (per column), Pu = 450 kips
E10.2.
For the prestressed girder alternate of the Project Bridge, estimate the
expected, overstrength plastic shear of the pier. See Exercise E10.1 for
details of the pier. Revise the displacement capacity estimated in E10.1 to
account for the increased column axial loads during strong ground shaking.
E10.3.
For the prestressed, concrete girder alternative of the Project Bridge, estimate the shear resistance of the columns for the expected, overstrength
plastic shear seismic loading calculated in E10.2. See Exercises E10.1 and
E10.2 for additional details.
E10.4.
Suppose the project bridge is located at a site with the seismic data below.
The superstructure weight is estimated to be 9.4 kips per linear foot for the
concrete girder option, including two lanes of uniform live load. The column weights are 1.44 kips per linear foot. The pier cap and pier diaphragm
together weigh 145 kips. Abutment weights are 94 kips each, including the
abutment beam and backwall. Each integral, stub abutment is supported by
six piles with an estimated lateral stiffness of 20 kip per inch per pile. The
two 42-inch diameter columns at the pier are assumed to behave as cantilevers in the longitudinal direction with clear height equal to 29 feet. No passive resistance at the abutments is to be relied upon for seismic resistance.
a) Determine the seismic displacement in the longitudinal direction
using a simplified linear response spectrum analysis. Apply the shortperiod displacement amplification, Rd, if applicable (see Equation 10.5).
Cracked properties of the concrete columns may be taken to be equal to
35% of gross properties.
b) Determine the longitudinal seismic shear demand on the pier columns
from the simplified response spectrum analysis using a response modification factor R = 3.5.
341
Seismic Design of Bridges
c) Using an estimated yield curvature, ϕy = 0.000111 in−1, and ultimate
curvature, ϕu = 0.001390 in−1, estimate yield and ultimate displacement
for the pier in the longitudinal direction. Other pertinent data may be
taken from Exercise 10.1.
d) Determine the ductility demand on the pier in the longitudinal direction.
e) Using an appropriate bi-linear load-deflection relationship, apply a
substitute-structure method analysis to obtain a second estimate of the
longitudinal displacement. Assume that the abutments remain linear
and experience no damage.
f) Given that the abutments have been assumed to remain in the linear
elastic range of behavior, determine the required shear resistance for
abutment piles.
g) Select and scale a 14-record ground motion suite appropriate for use in
a non-linear response history analysis of the Project Bridge using the
design response spectrum as the target response spectrum with a target
standard deviation (natural logarithm-based) of 0.6 across all periods.
The modal magnitude for the project site is Mw = 7.6. Near-field records
are not considered appropriate for the Site Class D project location. Use
the natural period computed in the previous step to establish a rational
period range of interest for ground motion scaling.
AS = 0.644
S DS = 0.927
S D1 = 0.863
TL = 8 seconds
11
Seismic Isolation
of Bridges
Properties of isolation bearings, including both lead-rubber bearings (LRB) and
friction-pendulum systems (FPS) were discussed and summarized in Section 8.3.
In the current chapter, preliminary design considerations, based on partial isolation, are presented for a constructed bridge on Interstate 40 over State Route 5 in
Madison County, Tennessee. In partial isolation for this structure, isolation devices
were incorporated into the pier, with abutments remaining integral. Also included is
the ground motion selection and modification which was adopted for the final design
of the bearings. The preliminary design using the substitute-structure method (SSM)
presented here (see Section 10.5) resulted in bearing specifications which required
no modification when the inelastic response history analysis was performed. That is
to say, the simplified, hand-calculated solution using a modified substitute-structure
type of analysis, accurately predicted inelastic response history results.
Also presented are simple, preliminary calculations for the Hernando De Soto
Bridge carrying Interstate 40 over the Mississippi River in Memphis, Tennessee. The
arch bridge has been retrofitted with FPS devices.
11.1 PARTIAL ISOLATION OF INTERSTATE
40 OVER STATE ROUTE 5
Modern seismic design of bridges typically requires that substructures be designed
for the over-strength plastic shear of the columns. For the subject bridge, over-sized
piers were used for aesthetic reasons. Design of foundations and pier caps for the
plastic shear of over-sized columns can require dramatic cost increases for the substructures – piling, pile caps, shafts, etc. With continued emphasis upon the seismic hazard in the New Madrid Seismic Zone (NMSZ), non-traditional techniques,
like isolation, may need to be considered in an increasing number of cases. For this
design, a three-span bridge in Madison County, Tennessee was analyzed for both
non-isolated and partially isolated conditions with the goal of reducing the demand
on piling and pile caps to at least partially offset the cost of the isolators. Partial
isolation offers the benefit of elastic substructures at the piers without costly expansion joints at the abutments. The final design for this structure included the partial
isolation alternative.
The subject bridge consists of three continuous, steel I-girder spans (95 ft, 156 ft,
95 ft long) and is 129.25 feet wide. The superstructure consists of fifteen welded steel
plate girders and a composite cast-in-place concrete deck with 3 ft high traffic parapets and a design allowance for a 3.5 inch asphalt overlay. The total superstructure
DOI: 10.1201/9781003265467-11
343
344
LRFD Bridge Design
weight is 28.6 kips per foot with a center of gravity 4.50 feet above the top of the
pier cap. Integral abutments on friction pipe piles are located at each end of the
bridge. All substructures are skewed 4.9 degrees from normal. An isometric model
of the structure is given in Figure 11.1. The cross section of the bridge is shown in
Figure 11.2.
Table 11.1 summarizes seismic design data for the design basis event (DBE) and
for the maximum considered event (MCE). The hazard levels are taken to correspond to the 7% probability of exceedance (PE) in 75 years and the 3% probability
of exceedance (PE) in 75 years, respectively. Whereas the 7% PE in 75-years ground
shaking is specified for design in the LRFD BDS, the 3% PE in 75-years ground
shaking was used for design of the isolation system on this project.
Passive resistance from backfill at the abutments was conservatively neglected for
both non-isolated and isolated conditions. Abutment piles were specified as 20 inch
diameter × 5/8 inches thick since anticipated displacement demands were greater
than 4 inches. The LRFD GS, in Section 5.2.4.2, requires that smaller diameter piles
be ignored for lateral stiffness in such cases:
“For pile-supported abutment foundations, the stiffness contribution of piles less than
or equal to 18 in. in diameter or width shall be ignored if the abutment displacement
is greater than 4 in., unless a displacement capacity analysis of the piles is performed,
and the piles are shown to be capable of accommodating the demands.”
The abutment pile cross-section D/t-ratio qualifies as ductile. This was judged necessary to prevent local buckling of the abutment piles during strong ground shaking.
FIGURE 11.1
Interstate 40 over State Route 5 – Madison County, Tennessee.
FIGURE 11.2
Interstate 40 cross section.
345
Seismic Isolation of Bridges
TABLE 11.1
Seismic Design Parameters for Interstate 40 over State Route 5
Parameter
DBE Hazard-level
MCE Hazard-level
0.279
0.482
SS
S1
AS
SDS
SD1
TS
TO
T*
Latitude
Longitude
0.554
0.148
0.347
0.626
0.411
0.656
0.131
0.820
35°40'14" N
88°49'47" W
0.952
0.269
0.491
0.917
0.655
0.714
0.143
0.893
35°40'14" N
88°49'47" W
Site Class
D
D
PGA
The natural periods of the non-isolated bridge are 0.21 seconds (transverse direction) and 0.76 seconds (longitudinal direction). The pier design basis, for either the
1,000-yr or the 2,500-yr ground motion, is the over-strength plastic shear, VPO. For
even the minimum amount of reinforcement in the columns, VPO was estimated to
be 8,044 kips per pier. This is the problematic feature of the subject bridge – due
to the width of the structure combined with the over-sized columns, even the minimum reinforcement permitted by the LRFD BDS results in extremely large seismic
shears. The pile caps and piling must be designed to remain elastic when subjected to
the large over-strength shears and moments at the base of the columns, for a design
without the isolation bearings.
Ground motion record pairs were selected from appropriate magnitude events and
Site Class D or E conditions at the recording station. Each record pair was scaled to
match the target spectrum in the period range of 0.20–4.00 seconds by minimizing
the mean square error between each individual record pair and the MCE target spectrum. These record pairs were then used for the nonlinear response history analysis
(NLRHA) of the isolated bridge. Figure 11.3 includes the plot of the ground motion
average spectra for comparison to the target spectrum at the MCE-level of ground
shaking. Table 11.2 summarizes the 14-record suite. A 14-record suite is twice the
minimum of seven required in AASHTO. Four of the records (nos. 11–14) are synthetic records developed in published research specifically for the NMSZ (Atkinson
and Beresnev, 2002, Fernández, 2007). Since only the mean response was to be
determined, no target, log-based standard deviation was specified. This would likely
be a needed change for future performance-based design procedures.
For the selected record suite, the minimum suite mean-to-target PSA ratio in
the period range of interest is 0.913 (greater than 0.900), the average ratio is 1.047
(greater than 1.000), and the maximum ratio is 1.177 (less than 1.300). The ground
346
LRFD Bridge Design
FIGURE 11.3
Target and suite mean PSA.
TABLE 11.2
Ground Motion Record Metadata
Year
Station
MW
R
km
VS30
m/s
Site Class
f
1
Tabas, Iran
1978
Boshrooyeh
7.35
24.1
338
D
3.84
2
3
4
5
6
7
8
9
10
11
12
13
Taiwan SMART1(45)
Landers
Landers
Kocaeli, Turkey
Chi-Chi, Taiwan
Chi-Chi, Taiwan
Sierra el Mayor
Darfield, NZ
Darfield, NZ
Atkinson-Beresnev
GaTech-Fernández
GaTech-Fernández
1986
1992
1992
1999
1999
1999
2010
2010
2010
2001
2006
2006
I01
Amboy
Yermo FS
Bursa Tofas
CHY036
HWA048
M5057
HORC
REHS
Memphis
Paducah
Paducah
7.30
7.28
7.28
7.51
7.62
7.62
7.20
7.10
7.10
8.00
7.70
7.70
56.0
69.0
23.6
60.0
16.1
47.4
41.0
7.0
19.0
53.8
25.9
25.9
274
271
354
275
233
278
163
326
141
205
295
295
D
D
D
D
D
D
E
D
E
D
D
D
2.37
3.01
2.18
3.55
1.49
2.56
2.32
0.96
1.67
1.44
1.35
1.18
14
GaTech-Fernández
2006
Jonesboro
7.70
30.4
205
D
1.55
No.
Event
Note: Records 11–14 are synthetic records from the literature.
347
Seismic Isolation of Bridges
motion suite was thus deemed appropriate for design of the isolators based on mean
response for the inelastic response history analyses (14 record pairs; 14 demand
values).
Table 11.3 is a summary of the ground motion parameters corresponding to the
14-record-pair suite.
Effective damping and stiffness properties were adopted for a substitute-structure-method (SSM), simplified response spectrum analysis. This is the method first
proposed by Gulkan and Sozen (Gulkan and Sozen, 1974), further developed by
Priestley and others (Priestley et al., 2007), and the basis for the simplified analysis
procedures in the AASHTO GS ISO (AASHTO, 2014).
Effective stiffness was taken to be equal to the secant stiffness. Effective damping is imparted to the system through hysteretic behavior of yielding elements. An
isolation device may be completely defined by the three parameters: (a) characteristic strength, Qd, (b) post-yield stiffness, Kd, and (c) post-yield stiffness ratio, α = Kd /
Ki, where Ki is the initial stiffness of the isolator. The isolator yield displacement, Dy,
is related to α, Qd, and Kd as given by Equation 11.1.
For the analysis of the LRB isolators on this project, α was taken to be equal to
0.10. The initial stiffness is due to the lead plug and elastomer stiffness values in
parallel with one another. Once the lead plug has yielded in shear, only the stiffness
contribution from the elastomer remains. The true yield strength is related to the
characteristic strength by Equation 11.2.
Rather than computing the effective stiffness of the isolators at a given pier, it is
convenient to recognize that the isolator-pier assembly in series results in a bilinear
behavior as depicted in Figure 11.4. The composite stiffness values, KSUB1 and KSUB2,
depend on the isolator parameters – Kd and α – and on the pier stiffness, KPIER, and
can be shown to be given by Equations 11.3 and 11.4.
TABLE 11.3
Suite Ground Motion Parameters
Parameter
MCE Record Set
Max acceleration (g)
0.397
Max velocity (cm/sec)
Max displacement (cm)
5 × Vmax/Amax (sec)
8 × Dmax/Vmax (sec)
Acceleration RMS (g)
Velocity RMS (cm/sec)
Displacement RMS (cm)
Arias intensity (m/sec)
Specific energy density (cm2/sec)
Cum. abs. velocity (cm/sec)
62.9
36.1
0.81
6.89
0.063
13.5
10.0
4.03
12,035
2,412
Acc spectrum intensity (g-sec)
0.364
348
FIGURE 11.4
LRFD Bridge Design
Bi-linear load displacement.
The real power of this direct displacement-based design procedure is apparent
in that the engineer decides upfront the desired values for V TARGET and ΔTARGET, the
shear and displacement, respectively, at each pier under seismic loading.
It is suggested here that isolators at each pier be designed such that V TARGET and
ΔTARGET are identical for each pier, resulting in a rigid body translation of the deck
with minimal torsional effects from eccentricity of mass with respect to the center
of strength or stiffness. One-half of the smallest yield shear for any of the piers is
suggested as an approximate value for V TARGET. Higher mode effects from pier local
vibration modes will increase the total shear on the pier and are, in many cases, of
the same order as the isolation mode shear, V TARGET; hence, the reason for starting
with such a small value for V TARGET. It will be convenient to express Qd in terms of the
other variables. Given target shears and displacements and preliminary values for
isolator post-yield parameters, the resulting expression is given by Equation 11.5. For
cases in which it is more desirable to assign a preliminary value for Qd and calculate
the necessary Kd, Equation 11.6 is provided.
Composite values for post-yield stiffness ratio, αSUB, and displacement ductility,
μSUB, are given in Equations 11.7 and 11.8, respectively, and are used to compute the
effective damping at each substructure. This is the slight deviation from the procedure described in AASHTO (AASHTO, 2014). The effective damping (ξEFF ) for
each pier is determined from Equation 11.9. Since a uniform displacement profile is
349
Seismic Isolation of Bridges
targeted, the total system damping is computed from Equation 11.10. The damping
response modifier (BL = 1/Rξ) is given by Equation 11.11. In Equation 11.10, K ABUT is
the elastic stiffness of the abutment, taken to be equal to the abutment pile stiffness
only for this project, with no contribution from backfill. The suggested value for
ξABUT, the elastic damping of the abutments, is 0.05.
It is further assumed that the design response spectrum is inversely proportional
to the effective period so that Equations 11.12 and 11.13 are valid. In other words, the
assumption is made that the effective period is between TS and TL, with verification
after the fact. This will typically be the case.
Here, SD is the spectral displacement (inches) at TEFF, SA is the spectral acceleration (g) at TEFF, and SD1 is the spectral acceleration (g) at a period of 1 second. The
spectral displacement thus computed is the total displacement of the mass. This is
compared to an initially assumed displacement. Once the desired accuracy has been
attained, the iterations are stopped and the design displacement has been determined.
A target, unidirectional deck displacement of 6 inches was selected to derive trial
isolator properties using the simplified approach. Upon iteration for the MCE-level
response spectrum, the converged deck displacement was found to be 5.60 inches.
The characteristic strength was selected to ensure that the isolator lead plugs do not
yield under Strength limit state loads (wind, braking, live, thermal, etc.). The simplified analysis is a unidirectional analysis. Bidirectional effects were approximated in
the simplified analysis by assuming:
1. The maximum component is equal to 1.3 times the geometric mean of two
components and
2. When one component (transverse or longitudinal) is at its maximum, the
perpendicular component is at 60% of its maximum.
This is a departure from the practice typically taken in design offices today.
Justification for the method may be found in the literature (Huff, 2016b). The bidirectional displacement demand from the simplified analysis was determined to be
8.26 inches, including thermal offset effects. These required properties of the isolators from preliminary design are listed in Table 11.4. Multiple LRB manufacturers’
TABLE 11.4
Preliminary LRB Isolator
Design from Simplified Analysis
Substructure
Qd
k/beam
kd
k/in/beam
Abutment 1
Integral
Integral
15.8
15.8
5.7
5.7
Integral
Integral
Pier 1
Pier 2
Abutment 2
350
LRFD Bridge Design
catalogs (FIP Industriale, 2011, Maurer, 2011, Robinson Seismic Ltd., 2011) provide
isolators capable of displacing more than 8.26 inches while carrying vertical loads
comparable to those expected at the piers of the subject structure.
Preliminary design with extremely stiff substructures may be carried out following procedures for the SSM outlined in Chapter 10, Section 10.4.
Dy =
Qd a
×
Kd 1 - a
(11.1)
Qd
1-a
(11.2)
Fy =
K SUB1 =
K PIER K d
a K PIER + K d
(11.3)
K PIER K d
K PIER + K d
(11.4)
K SUB 2 =
æ
Kd
Qd = VTARGET ç 1 +
K PIER
è
Kd =
VTARGET - Qd
V
DTARGET - TARGET
K PIER
(11.5)
(11.6)
a K PIER + K d
K PIER + K d
(11.7)
DTARGET DTARGET (1 - a ) K PIER K d
=
DYSUB
Qd (a K PIER + K d )
(11.8)
2 ( mSUB - 1) (1 - a SUB )
pmSUB (1 + a SUB mSUB - a SUB )
(11.9)
a SUB =
mSUB =
ö
÷ - DTARGET K d
ø
x SUB =
x EFF =
2 K ABUT
æ
VTARGET ö
ç x SUB D
÷
TARGET ø
è
æ VTARGET ö
ç
÷
PIERS è D TARGET ø
å
+å
2 K ABUT x ABUT +
PIERS
(11.10)
351
Seismic Isolation of Bridges
BL =
TEFF = 2p
SD =
1 æ x EFF ö
=
Rx çè 0.05 ÷ø
0.30
£ 1.7
W
é
g ê2 K ABUT +
ë
S D1 g
×
TEFF BL
x* = x
å
æ VTARGET ö ù
ç
÷ú
PIERS è D TARGET ø
û
(11.11)
(11.12)
2
g S D1
æT ö
× ç EFF ÷ =
×
× TEFF
2
BL
2
p
4
p
è
ø
1 - 0.1 ( m - 1) (1 - a )
m
1 + am - a
(11.13)
(11.14)
Nonlinear modal response history analysis (NLRHA) – Fast Nonlinear Analysis
(FNA) –methods were used to verify the simplified SSM design. These procedures
are extremely efficient and accurate for problems in which nonlinear behavior is confined to link-type elements. CSiBridge (Computers and Structures, Inc., 2016) was
used for the analysis of the bridge. Nonlinear link elements in the CSiBridge library
include biaxial hysteretic elements with coupled plasticity for the two shear deformations. The model incorporated into CSiBridge is that proposed by Wen (Wen, 1976)
and recommended for isolators by Tsopelas et al. (1991). The isolators were modeled
with non-linear properties in both shear directions.
To capture the most accurate load transfer into the non-linear link elements, a
finite element model of the bridge was created. The bridge superstructure was modeled using a mixed use of frame and shell elements to best represent the components
of the welded plate girder and composite slab. Pipe pile supports at the abutments
were modeled as elastic springs at each girder having full vertical restraint. The
piers were modeled as non-prismatic frame elements with all column bases fixed for
translation and rotation. The girder bases were connected to the pier cap using the
non-linear links defined for the isolators.
Ritz vectors were used instead of the usual eigenvector modal analysis. Ritz vectors are generally preferred for this type of analysis (FNA) unless every possible
mode is included in the eigenvector analysis.
Elastic damping was taken as 2.5% of critical for the response history analyses.
This small damping value was adopted due to problems which can frequently occur
with viscous damping models in nonlinear analyses. It has been shown (Priestley
and Grant, 2005) that the appropriate elastic damping for nonlinear analyses is significantly less than the typical assumed value of 5%. Suggested values (Priestley
et al., 2007) developed in the referenced works are repeated here in Equation 11.14.
352
LRFD Bridge Design
For LRB systems, α = 0.10 and the above becomes zero when μ = 12.111. So, at
least for the case of Rayleigh damping, it would appear that the choice of damping
less than 5% of critical is appropriate for the nonlinear analysis of isolated structures given that it is not at all uncommon for isolated systems to have μ values well
above 12.
Final design isolator demands – taken as the average from the 14 NLRHA results
at each hazard level – are summarized in Table 11.5.
The correct means of calculating the isolator demand for a load case is to compute
the resultant displacement at each time step and find the maximum for that case.
Undue conservatism would result if the maximum longitudinal demand were combined with the maximum transverse demand since the two do not generally occur
simultaneously. Clearly, the simplified preliminary design procedure proved reliable
in this case. This can be expected when the deck aspect ratio is such that the superstructure behaves very nearly as a rigid block.
A key parameter in determining the economy of an isolation system for bridges
is the shear transmitted to the substructures. Table 11.6 summarizes these values for
both simplified analysis and the NLRHA.
Partial isolation resulted in a significant decrease (89% decrease, relative to overstrength plastic shear) in the load transmitted to the substructures at the intermediate piers, compared to the non-isolated conventional design strategy. This does
carry along an increase in the load transferred to the abutment piling. However, the
nominal shear resistance of the 20 inch diameter × 5/8 inch thick pipe piles with a
minimum yield strength of 35 ksi is 356 kips per pile. The load transmitted to the
abutments during the 2,500-yr ground shaking was estimated to be 155 kips per pile,
about 44% of the nominal capacity.
TABLE 11.5
Isolator Displacement Demands
SSM
MCE TDD
inches
NLRHA
MCE TDD
inches
Pier 1
8.02
7.85
Pier 2
8.02
7.85
Substructure
TABLE 11.6
Shear Transmitted to Piers
Condition
SSM
kips/pier
NLRHA
kips/pier
Non-isolated
8,044
8,044
Isolated
1,185
857
Seismic Isolation of Bridges
353
11.2 SEISMIC RETROFIT OF INTERSTATE 40
OVER THE MISSISSIPPI RIVER
The Hernando de Soto Bridge carries Interstate 40 across the Mississippi River near
Memphis, Tennessee. The bridge consists of two 900-ft tied arches across the river
with multiple approach spans on both sides of the main spans.
The bridge has been retrofitted with friction-pendulum system devices at the
three arch piers and LRB isolation bearings at approach spans.
Problem 11.1 describes some of the bearing details and the seismic analysis
results for the FPS devices.
11.3 SOLVED EXAMPLES
Problem 11.1
Properties for the friction-pendulum system bearings on Interstate 40 over
the Mississippi River are summarized below. Figures P11.1a and P11.1b
depict the design response spectra and the arch spans elevation, respectively. Perform a simple, substitute-structure analysis of the isolated bridge
as a sanity check on the reported isolator demands. Refer to Chapter 8,
Section 8.3, for equations defining the FPS properties. The SSM analysis is
discussed in Section 10.4 of Chapter 10. The deck width is 88 feet.
End Piers A and C:
• R = 244 inches, radius of concave surface
• μ = 0.06, dynamic friction coefficient
• DMax = 27.25 inches, from inelastic response history analysis
• W = 5,405 kips per bearing (2 bearings per Pier)
• D = 8 ft 10 inches, diameter of bearing
FIGURE P11.1A DRS for Problem P11.1.
FIGURE P11.1B
Elevation for Problem P11.1.
354
LRFD Bridge Design
355
Seismic Isolation of Bridges
Center Pier B:
• R = 244 inches, radius of concave surface
• μ = 0.06, dynamic friction coefficient
• DMax = 18.75 inches, from inelastic response history analysis
• W = 12,611 kips per bearing (two bearings per pier)
• D = 8 ft 10 inches, diameter of bearing
Estimate the maximum force exerted on the piers and the vertical displacements as well.
Assume that the given design response spectrum is a geometric meanbased design response spectrum.
PROBLEM 11.1
TEH
1/6
356
PROBLEM 11.1
LRFD Bridge Design
TEH
2/6
357
Seismic Isolation of Bridges
PROBLEM 11.1
TEH
3/6
358
PROBLEM 11.1
LRFD Bridge Design
TEH
4/6
359
Seismic Isolation of Bridges
PROBLEM 11.1
TEH
5/6
360
LRFD Bridge Design
PROBLEM 11.1
TEH
6/6
The SSM analysis provides a good sanity check on the final design results from
nonlinear response history analysis (NLRHA).
SSM results are slightly higher at the central pier, Pier B:
• 18.75 inches from NLRHA
• 22.97 inches from SSM
SSM results are very close, but slightly lower at the end piers, Piers A and C:
• 27.25 inches from NLRHA
• 26.97 inches from SSM
11.4 EXERCISES
E11.1.
Perform the transverse direction substitute-structure analysis of partially
isolated Interstate 40 over State Route 5 (Figures 11.1 and 11.2). A summary
of design parameters is given below.
AS = 0.491 g
SDS = 0.917 g
SD1 = 0.655 g
Steel Beams (15):
R DL = 275 k/beam
R LL = 56 k/beam
Isolators:
Qd = 15.8 k/beam
kd = 5.7 k/in/beam
Δnon-eq = 0.75 inches, non-seismic displacement demand
Superstructure:
Wss = 29.6 kips/ft, includes six lanes of HL-93
uniform load
Integral abutments: Wabut = 427 kips, each
21 pipe piles per abutment, Kpile = 20 kips/inch/pile
Seismic Isolation of Bridges
Spans:
Piers/Bents:
95 ft, 156 ft, 95 ft = 346 ft total bridge length
KPier = 18,106 kips/inch/pier, transverse
KPier = 640 kips/inch/pier, longitudinal
ϕVy = 0.9Vy = 5,171 kips/pier, transverse
ϕVy = 0.9Vy = 1,097 kips/pier, longitudinal
Vpo = 8,359 kips/pier, transverse (overstrength, plastic)
Vpo = 1,748 kips/pier, longitudinal (overstrength, plastic)
WHM = 479 kips/pier, lumped mass for higher mode effects
Vu = 101 kips/pier, Strength limit state shear
361
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Index
AASHTO, 1, 6, 23, 34, 63, 83, 107, 125, 179, 215,
241, 305
AASHTO Bulb-T, 9, 11, 12
AASHTO I-Girder, 10–12
Abutment, 6, 27, 32, 66, 108, 220, 254, 266
Alaska, 258
Amplitude scaling, 313–316
Anchorage reinforcement, 250
Anchor rod, 13, 67, 217, 227
Arch bridge, 343
Arching, 190
Arias intensity, 347
Artificial ground motion, 315
ASCE 7-16, 33, 305, 314, 319
Aspect ratio, 352
ASTM, 10, 12–13, 65, 67, 139, 227, 249, 258
Average daily truck traffic (ADTT), 136
Average response, 316
Axial force, 179, 183, 244, 312
Axial resistance, 133–134, 245, 259
Axis of rotation, 215
Axle load, 25–27, 87
Backfill, 266, 310–311, 344, 349
Backwall, 266, 310
Barrier, 29–31, 83, 109
Base metal, 136
Beam-slab bridge, 112
Bearing design, 2, 218
Bearing pressure, 221
Bearing resistance, 133–134
Bearing stiffener, 125, 133–134
Bearing stress, 221
Bent cap, 241, 309, 313
Bent column, 9, 31, 241–248, 254–260, 265,
305–312, 320, 343, 351
Biaxial flexure, 246
Bidirectional ground motion, 349
Bi-linear, 317–318, 348
Blast load, 64
Block shear, 67, 140, 142
Bolt, high strength, 13, 65, 67, 125, 138, 141–142
Bolt hole, 140, 142
Bolt tension, 141
Bonded area, 223
Bonded plate, 219
Bonded reinforcement, 180, 181
Bottom mat, 85, 187
Braking force, 23, 27, 225
B-region, 182, 243
BridgeLink, 14
Bridge loads, 23, 267
Bronze plate, 221
Buckling, 128–129, 144–146, 245, 260, 262,
264–265, 344
Bulb-T beam, 9–12, 179
Bulge, 216, 225
Camber, 125
Candidate records, 313–314
Capacity design, 262, 307, 311–313
Capacity protection, 306–308
Cap soffit, 258
Cast-in-place, 3, 5, 83, 110, 179, 183–184, 188,
241, 258, 343
Center Hill Lake, 220–221
Centrifugal force, 23, 27–28, 64, 111
Characteristic magnitude, 314
Characteristic strength, 225, 347, 349
Circular bearing, 215, 218, 223, 225
Clay, 35, 251–253, 310
Clay County, 215
Clear cover, 87–88
Closed-end, 249
Closed-form, 261, 262
Coefficient of friction, 29, 221
Cohesionless, 35, 310
Collision, vehicle, 30, 63
Collision, vessel, 31, 63–64
Column base, 351
Compact section, 125–126
Composite beam, 182, 190
Composite properties, 7
Compression-control, 65, 67–68, 188–189,
241–242
Compression fiber, 126, 144, 182, 188, 242–243
Compression flange, 125–132, 142, 144–145
Compression reinforcement, 88, 180, 188
Compressive stress block, 188
Computer modeling, 255
Concave surface, 222, 226–227, 353
Concentrated load, 183, 244, 247
Concrete fill, 249, 252, 258
Confining stress, 256
Conjugate beam, 16
Connection design, 186
Connection plate, 132–133
Constitutive model, 262
Constructability limit state, 125, 131
Construction sequence, 266
367
368
Contact surface, 221
Contingency, 33, 225, 264
Continuity, 3, 7, 179, 184–190
Continuity diaphragm, 185
Contractor, 3, 187, 221
Contraflexure, 26, 108, 309
Controlling flange, 138
Control point, 34–35, 305
Conventional design, 24, 305, 316, 352
Copper plate, 221
Core area, 254, 307
Core diameter, 254, 307, 321
Correction factor, 107, 110
Crack control, 63, 87, 189, 243
Cracked section, 189, 266
Cracking moment, 185
Crash force, 85
Creep, 7, 63, 266
Critical load, 245
Critical section, 247, 256
Cross-frame, 3, 7, 13, 24, 112, 125, 128–132,
142–146, 305
CSiBridge, 351
Curvature, 27, 111, 143, 145, 227, 245, 257
Cyclic component, 225
Cyclic stress, 134
Damage control, 255
Damping, 225, 308, 318, 347–352
Debonding, 190
Deck, 1, 24, 83, 190, 243, 265, 266, 310, 343, 348
Deck overhang, 5, 83–87
Deflection, 14, 179, 253
Deformation-related, 64
Depth of web in compression, 131
Depth to fixity, 252–253, 264
Design response spectrum, 34–35, 305, 308,
317–318, 349
Design section, 86
Design speed, 27, 29
Design tandem, 25–27
Design truck, 25–29, 83, 111
De-tensioning, 179
Development length, 185–186, 190
Diaphragm, 2–3, 7, 13, 24, 112, 132, 143, 145,
179, 184, 188, 227
Displacement amplification factor, 308, 319
Displacement-based design, 42, 255, 305,
308, 348
Displacement capacity, 255, 257, 308
Displacement ductility, 308, 316, 318, 321, 348
Distribution factor, live load, 28–29, 107–112
Distribution reinforcement, 83
Double curvature, 143, 145, 245
Dowel, 250, 264
Drag coefficient, 29, 45
Index
Draping, 190–191
D-region, 182, 243
Drilled shaft, 241–242, 250–251, 258
Dual bridge, 46
Ductility, 24, 64, 254, 308, 316–318, 348
Ductility factor, 24
Dynamic friction coefficient, 226, 353
Dynamic load allowance, 23, 64
Earth pressure, 23, 64
Eccentricity, 112, 179, 246–247, 319, 348
Effective column section, 134
Effective damping, 225, 229, 318, 347–348
Effective length, 133, 254
Effectiveness coefficient, 256
Effective period, 316–318, 349
Effective stiffness, 347
Effective strip width, 83, 87
Eigenvector, 351
Elastic seismic force, 306–307
Elastic shortening, 180
Elastic substructure, 305, 343
Elastomer, 33, 215–216, 222, 347
Elastomeric bearing pad, 215
End panel, 132, 146
Engineering, News, Record (ENR), 250, 266
Epoxy-coated, 87, 187
Essentially elastic, 305, 308, 311
Euler, 260
Excel, 13, 137
Expansion abutment, 254, 325, 343
Expansion bearing, 215, 217–218, 220–221
Expansion joint, 27, 254, 306, 310, 343
Expected strength, 254, 258, 262, 307, 309, 311
Exposure, 253
Exposure coefficient, 29
Exposure factor, 87, 189, 242
Exposure time, 33
Extended strands, 185–186
Exterior girder, 5, 26, 83, 85, 109–112
External confinement, 258
Extreme Event limit state, 23, 63–66, 85, 148,
247, 261, 263, 320
Fast nonlinear analysis (FNA), 351
Fatigue, 134–140, 187, 189, 265–266
Fatigue category, 134, 136
Fatigue limit state, 25–27, 63, 66, 107–108, 111,
135, 187, 190, 218
Fatigue resistance, 136, 146–147
Fatigue threshold, 135, 216, 266
Fatigue truck, 25, 111
Fault-normal, 316
Fault-parallel, 316
Federal Highway Administration (FHWA), 13,
142, 316
369
Index
Fiber, 5, 126, 144, 148, 182, 188, 242–243, 264
Field splice, 3, 13, 125, 131, 137
Filler, 24
Filler plate, 137–139
Finite element, 351
Finite-life, 63, 135–136
FIP Industriale, 350
First-order, 128–129, 309
Fixed bearing, 217–221
Fixity, 252–253, 264
Flange, 13, 86, 125–137, 142–146, 179–180,
186–188
Flange splice plate, 138–140
Flexural resistance, 84, 87, 126, 138–139,
179–180, 188, 260–262
FNA, see Fast nonlinear analysis
Footing, 241–242, 246–247, 309–313
Force-based, 42, 255, 305–306, 311
Force reduction factor (r), 306–307
Force-related, 64
Foundations, 343–344
Fourier spectrum, 315
Frame elements, 351
Frequency content, 314
Frequency domain, 314–315
Friction, 23, 29, 221–223, 226, 343–344, 353
Frictional force, 23, 64, 221
Friction-pendulum, system, (FPS), 222, 343, 353
Friction pile, 266, 344
Gates method, 250, 266
Geometric mean, 33–34, 305, 349
Girder, prestressed concrete, 1, 7–8, 179
Girder age, 185–186
Girder spacing, 5–6, 14, 83, 86, 107, 145
Girder system stiffness, 144–146
Global system buckling, 146
Gross section, 67, 140, 142
Ground motion, 23, 33–34, 263, 306, 318,
345, 347
Ground motion database, 314–315
Ground motion modification, 314, 343
Ground motion records, 53, 305, 313–314, 323,
345–346
Ground motion selection, 305, 313, 316–317, 343
Ground motion suite, 305, 313–317, 345–347
Gulkan, P., 317, 347
Gust, 29
Harping, 191
Haunch, 24
Hernando De Soto Bridge, 223–224, 343, 353
High-performance steel (HPS), 12
Hinge, 252, 256, 258, 264, 305, 309, 312, 320
Hinge region, 253–254, 307, 320–321
HL-93 live load, 25–27, 87
Hold-down, 191
Hole-related factor, 141
Homogeneous, 126, 128, 131
Hoop, 254–257, 307, 312, 321
Horizontal Shear, 183
H-pile, 248
Hydrostatic stress, 216, 249
Hysteretic, 318, 347, 351
I-beam, 6, 8–12, 179
Impact, 23, 25–26, 31, 85, 87, 136, 265
Implicit displacement capacity, 256
Inelastic behavior, 225, 263, 308
Inelastic response history analysis, 305–306, 308,
343, 347, 353, 355
Infinite-life, 63, 135–136, 190, 265
Inflection point, 143
In-ground hinge, 253, 258
Initial prestress, 187
Initial strength, 21
Inner splice plate, 176
Intact rock, 250
Integral abutment, 27, 254, 266–267, 344
Interface shear, 183–184
Interior bay, 85
Interior girder, 108–111
Intermediate panel, 132
Interpolation, 42, 68, 188–189, 241, 246
Interstate-40, 222, 253, 343–345
Isolation, seismic, 215, 222, 305, 318, 343
Isolation bearing, 215, 222, 225, 305, 343,
345, 353
Jetting, 249
Joint reinforcement, 258
Joint shear, 247
K-frame, 176
Lane, 26, 107
Lane loading, 25–26, 87
Lateral brace, 146
Lateral drag coefficient, 45
Lateral flange bending, 127–129
Lateral torsional buckling, 129, 144
Lead plug, 222–223, 347, 349
Lead-rubber, bearing, (LRB), 222, 225, 227, 343,
347, 349, 352–353
LEAP, 2, 13
Lever rule, 29, 107–111, 113
Limit state, 63
Linear interpolation, 68, 188–189, 241, 246
Line-girder analysis, 5, 107
Link (element type), 351
Live load, 25–27
Live load distribution factor, 28–29, 107
370
Load combination, 63–66
Load factor, 24, 63–66, 85, 86
Load modifier, 24
Local buckling, 129, 262, 264–265, 344
Longitudinal axis, 71, 146, 182, 243
Longitudinal drag coefficient, 45
Longitudinal mat, 88
Longitudinal stiffener, 126, 132
Long-term composite, 7, 69, 126, 128
Loss of prestress, 9
Lower limit, 183, 227, 244
Lowlands, 36–39
Low-relaxation, 9–10, 179
Low-temperature grade, 217
LRFD-Simon, 13, 124, 148–149, 174
Madison County, 222, 343–344
Mander model, 255, 312
Mass, 324, 348–349
Mass center, 319
Mass participation, 316, 324
Maurer, 350
Maximal, 217
Maximum considered event (MCE), 344–352
Maximum direction, 33, 40, 305
Maximum response, 316
MCE, see Maximum considered event
Mean recurrence interval, 33, 264
Mean-square-error (MSE), 313
Memphis, 222–224, 343, 346, 353
Metadata, 346
Metal deck forms, 148
Minimal, 217, 315, 348
Mississippi Embayment, 36, 40–44
Mississippi River, 222, 343, 353
Modal, 55, 351
Modal analysis, 324, 351
Modal magnitude, 324, 341
Modification factor (LTB), 144, 178
Modular ratio, 7, 14, 146, 260
Moment arm, 139
Moment gradient factor, 129
Moment magnitude, 315
Moment resistance, 9, 126, 138–139
Moment-to-shear (M/V) ratio, 199–203
Multi-presence factor, 25–27, 85, 87, 107–108,
111–112, 136
National Earthquake Hazards Reduction
Program (NEHRP), 40–45, 306, 313
Natural period, 341, 345
Near-fault, 315–316
Negative moment, 3, 5, 8, 26–27, 84–88, 108,
148, 179, 182, 184, 187
Net section, 67, 136, 140, 142
New Madrid seismic zone, 36, 40, 343
Index
NIST-GCR-11-917-15, 313
NLRHA, see Nonlinear response history analysis
Nominal resistance, 128, 139, 180, 182, 190, 243
Non-composite, 3, 7, 107, 126–129, 139, 141, 148,
180, 187
Non-ductile, 24, 307
Non-hybrid, 128
Nonlinear response history analysis (NLRHA),
345, 351–352, 360
One-way shear, 247
Orthogonal, 306, 308, 315–316
Oscillator, 318
Outer splice plate, 138, 141, 176
Over-size, 141, 343, 345
Over-strength, 311, 343, 345
Overturning, 312
Panel bracing, 142–143, 146
Parapet, 3, 5, 7, 31, 83–87, 107, 343
Partial isolation, 343, 352
Passive capacity, 310
Pedestrian, 23, 64
PEER, 314–315
Penalty factor, 139
Period range, 315–317, 345
Permanent load, 7, 23, 63, 181, 266, 307
Physics-based model, 315
Pier, 2–7, 26, 32, 43, 45, 133, 148, 227, 242, 245,
251, 255, 306, 343–353
Pile, 67, 241, 308–309
Pile bent, 251, 258, 306–307
Pile cap, 241, 247, 265, 313, 345
Pile driving, 67, 242, 250
Pile embedment, 252
Pile set, 250
Pile splicing, 252
Pipe pile, 249, 344, 351
Pitch (hoop), 253–254, 257, 307, 321
Pitch (shear stud), 146
Plastic displacement, 257
Plastic moment, 9, 14, 127, 148, 260, 309, 312
Plastic neutral axis, 126, 148
Plastic shear, 131, 307, 311–312, 343, 345, 352
Plastic stress distribution, 126, 259
Plate girder, 1, 12, 14, 136–137, 343, 351
Plunging, 266
POLA, 263–264
POLB, 263–264
Positive flexure, 125–129, 139, 148, 180
Positive moment, 3, 5, 21, 69, 84–85, 108,
127–128, 147, 185
Post-yield stiffness, 222–225, 347–348
Pot bearing, 215
Pouring sequence, 125
Precast panel, 187
Index
Predrilling, 249
Prestressed concrete beam, 181
Prestressed concrete pile, 253
Prestressing strand, 185, 191, 264
Prestress losses, see Loss of prestress
Presumptive pressure, 310
Priestley, M. J. N., 253, 255, 318, 322, 347, 351
Probability of exceedance, 33–34, 264, 344
Production pile, 267
Profile depth, 41
Project Bridge, 1, 6
Proportion limits, 126, 133
Pseudo-spectral acceleration (PSA), 34–35, 42,
314–319, 345, 346
PTFE, 215
Pulse-type ground motion, 314–318
Punching shear, 265
Pushover analysis, 255, 323
Radius of gyration, 134, 260
Random signal, 315
Re-centering, 227
Rectangular bearing, 215–216, 225
Recursive, 261
Redundancy, 24, 64
Regression, 86
Reinforced elastomer, 33, 216–217
Reinforcement (A 706), 9, 311
Resistance factor, 1, 23–24, 29, 63, 65–68, 87,
131, 139–144, 181–182, 188,
241–243, 250
Response 2000, 13, 68
Response history analysis, see NLRHA
Response modification factor (R), 340
Response spectrum, see Design response
spectrum
Restrainer, 221
Restraint moment, 185
Restraint system, 219
Resultant displacement, 352
Retrofit manual, 316
Rigid cross section, 107, 112
Rigid footing, 324
Rigid frame, 5, 254, 257, 309
Risk-targeted, 33, 305
Ritz vectors, 351
Robinson Seismic LTD, 350
rotation coefficient, 215
RotD100, 305, 314
RSCTH, 315
Rule-of-thumb, 15, 249
Rural interstate, 137
Sand, 251–252, 310
Sanity check, 353, 360
SD, see Spectral displacement
371
SDOF, see Single-degree-of-freedom
Second-order, 129, 245, 260, 309
Section analysis, 13, 68, 112, 244, 312
Section moduli, 21, 69, 177, 262
Section properties, 14, 110, 180, 189, 266
Seismic design category, 42, 43, 253, 257, 305,
309–311, 321
Seismic displacement, 222–225, 305, 317
Seismic hoop, 254, 307
Seismic isolation, see Isolation, seismic
SeismoArtif, 315
SeismoMatch, 315
self-weight, 7, 24, 87, 184, 248, 266
Serviceability, 255
Service limit state, 63–65, 87, 125, 179, 185–189,
216–217, 221, 242, 247, 249, 252, 266
Settlement, 23, 64
Shaft, see Drilled shaft
Shape factor, 215–216, 225
Shear deformation, 215, 217, 228, 351
Shear depth, 182, 243, 246, 265
Shear displacement, 215, 217
Shear length, 263
Shear modulus, 215, 217, 222, 228
Shear plane, 138–139, 184, 227
Shear range, 146
Shear resistance, 126, 131, 138–139, 182–184,
227, 243, 247, 260, 265, 320, 352
Shear strain, 215–218, 223, 225
Shear strain factor, 225
Shear stud, 13, 125, 146
Shear transfer, 183–184
Shear wave velocity, 34
Shell elements, 351
Short-term section, 7, 69, 128, 146
Shrinkage, 64, 266
Side bars, 242
Side resistance, 250–251
Sidesway, 245
SigmaSpectra, 314
Significant duration, 318
Simon, 13
Simplified design, 85, 87
Single-degree-of-freedom (SDOF), 318
Single-lane, 25–26, 108, 111, 136–137
Site class, 34–45, 345–346
Site factor, 35–45
Site response analysis, 35, 40
Skew, 6, 110, 131, 310, 344
Slenderness, 129, 180, 245, 254, 262, 265
Slider plate, 221, 228
Sligo bridge, 228
Slip-critical, 63, 138, 140
Slotted hole, 141–142
Socket, 250–251
Sozen, M., 317, 347
372
Spacing limit, 182, 243, 253
Specified strength, 185, 191
Spectral displacement (SD), 349, 351
Spectral matching, 314–315
Spectral shape, 313
Spiral, 67, 245, 253–257, 307, 312, 321
Spliced girder, 179
Splice plate, 138–142
SSM, see Substitute structure method
Stability bracing, 2, 128, 142
Stability check, 218–219
Standard deviation, 314, 317, 345
State Route-5, 222, 343–345
State Route-52, 215–216
State University of New York (SUNY), 315
Static component, 225
Steel assembly bearing, 220
Stiffener, 125, 126, 131–136, 145
Stiffness parameter, 107, 110
Stirrup, 182, 193, 243
Strain compatibility, 189, 259, 262
Strain hardening, 311
Strain limit, 67–68, 189, 228, 241, 263–264, 311
Strand, 7–10, 67, 179–187, 190–191
Strand release, 186
Stream pressure, 64
Strength limit state, 64–65, 83, 87, 125–134, 138,
146, 179, 185, 221, 246, 259, 266
Stress block, 180, 188
Stress modifier, 223
Stress range, 69, 131, 134–136, 190, 265
Strip width, 83–84, 87
Structural analysis, 13, 309–316
Substitute structure method (SSM), 317–319, 343,
347, 350–353
Substructure, 1, 6, 13, 27–32, 217–221, 241, 306,
309–311, 318, 343–344, 348
Suite, see Ground motion suite
SUNY, see State University of New York
Superelevation, 29
Support length, 310
Surface conditions (splice), 141–142
Surface-related factor, 141
Surface roughness, 29–30
Synthetic record, 315, 345–346
Target response spectrum, 315
Tectonic regime, 315
Temperature loading, 32
Temperature zone, 217
Tensile strength, 13, 21, 139, 147, 179, 227, 311
Tension-control, see Compression-control
Tension field action, 131
Tension flange, 126–132, 144–145
Test level, 31, 83
Thermal expansion, 32–33, 217, 220, 223
Thermal movement range, 217
Index
Tied arch, 353
Tip resistance, 250–251
Tolerance, 217, 249
Top mat, 85, 187
Torsional brace, 143–146
Torsional effects, 348
Traffic-induced, 216
Transfer length, 190
Transformed properties, 180
Transient load, 7, 23, 64, 181
Transition period, 35, 55
Transverse mat, 88
Transverse reinforcement, 83, 88, 253, 265, 308,
312, 321
Transverse stiffener, 131–133
Twin-girder system, 146
Two-way shear, 247, 265
Ultimate curvature, 257
Unbraced length, 143
Unfactored, 190, 265–266, 309
Unidirectional, 349
Uniform hazard, 33–34, 40, 305
United States Geological Survey (USGS), 34–36,
305–306, 314
Unstiffened, 131
Uplands, see Lowlands
Upper limit, 183, 227, 244
Upward component, 191
Urban interstate, 137
USGS, see United States Geological Survey
Variability, 313–314
Vector, 139, 351
Vertical displacement, 226
Vertical load capacity, 222
Vertical shear range, 146
VisualAnalysis, 13
Volumetric ratio, 264
Wearing surface, 23, 64
Weathering steel, 12, 136
Web distortion, 144, 176
Web splice, 138–140
Welded plate girder, 1, 12, 125, 343, 351
Weld metal, 67
Wheel load, 29, 83, 87
Wind load, 3, 29, 63–64, 128
Wrapping strand, 191
WSDOT, 258, 263, 265
WSDOT BridgeLink, 14
Yield curvature, 257
Yield force, 229, 324
Yield stress, 131, 148, 216, 223, 258, 266,
311, 321
Young’s Modulus, 7, 12, 107, 125, 259